The black bean is a small, shiny variety of the common bean (Phaseolus vulgaris) especially popular in Latin American cuisine, though it can also be found in the Cajun and Creole cuisines of south Louisiana. Like all varieties of the common bean, it is native to the Americas,[rx] but has been introduced around the world. It is also used in Indian cuisine, Tamil cuisine, where it is known as karuppu kaaramani and in Maharashtrian cuisine, where it is known as Kala Ghevada. The black turtle bean is often simply called the black bean (frijoles negros, zaragoza, judía negra, poroto negro, caraota negra, or habichuela negra in Spanish; and feijão preto in Portuguese), although this terminology can cause confusion with at least three other types of black beans.
Black Bean (Black Turtle Bean) Quick Facts | |
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Name: | Black Bean (Black Turtle Bean) |
Scientific Name: | Phaseolus vulgaris |
Origin | It is originated in Central and South America. |
Colors | Black |
Shapes | Small, oval-shaped |
Calories | 624 Kcal./cup |
Major nutrients | Copper (204.44%) Vitamin B9 (204.25%) Iron (200.13%) Vitamin B1 (138.00%) Phosphorus (115.71%) |
Health benefits | Prevention of cardiovascular problems, Source of antioxidants, High content of fiber, Digestive health, Provide energy |
Black bean is scientifically known as Phaseolus vulgaris (commonly called Black turtle bean, Dry bean), is a herbaceous and annual plant belonging to legume or bean family Fabaceae. It is originated in South and Central America and is cultivated widely in various parts of the world for its edible beans. The beans are harvested and consumed immaturely. The varieties of Phaseolus vulgaris varieties are developed in both bush and twinning forms. Bush forms reach 1 meter tall and grow in erect leafy clumps. Twining forms grow up to 4 meters long and need the support of poles or trellising. Plants possess trifoliate compound leaves having oval to rhombic leaflets measuring 16 cm long and are pubescent. Flowers are yellow, violet, white, or red which forms in loose and open unbranched clusters. Flowers are replaced by round to slightly flattened pods that measures about 15 cm long.
Black beans also known as turtle beans are classified as legumes. It has a hard shell-like appearance. It is a shiny variety of common beans which is well common in Latin American cuisine though it could be found in Creole and Cajun cuisines of South Louisiana. It is also used in Punjabi cuisine and is known as black beans. In the US, it is used correspondently with vigna mungo. Though it has been cooked and cultivated for thousands of years in South America, it did not reach the Atlantic till the 1500s in boats of European explorers. Similar to other legumes such as peas, peanuts, and lentils, Black beans are appreciated for their high content of protein and fiber. It contains various minerals and vitamins which are beneficial for human health. It is a great source of magnesium, thiamin, manganese, phosphorus, iron, and folate.
History
Black beans are native to Central and South America dating back 7000 years where they became a staple food. In the 15th century, it was discovered by Spanish explorers in Peru and was brought and introduced to Europe. Ultimately, it spread to Asia and Africa by Portuguese and Spanish traders and became popular for long-term storage ability, ease of growth, texture, taste, and nutrient. Presently, it is a staple diet in Cuba, Brazil, Mexico, Dominican Republic, and Guatemala cuisines. Brazil and India are the top leading producers of dried black beans. Indonesia, Mexico, and United States also produce these beans.
Nutritional value of Beans, black turtle, mature seeds, raw
Calories 624 Kcal. Calories from Fat 14.94 Kcal.
Proximity | Amount | % DV |
---|---|---|
Water | 20.24 g | N/D |
Energy | 624 Kcal | N/D |
Energy | 2609 kJ | N/D |
Protein | 39.1 g | 78.20% |
Total Fat (lipid) | 1.66 g | 4.74% |
Ash | 6.62 g | N/D |
Carbohydrate | 116.38 g | 89.52% |
Total dietary Fiber | 28.5 g | 75.00% |
Total Sugars | 3.9 g | N/D |
Minerals | Amount | % DV |
---|---|---|
Calcium, Ca | 294 mg | 29.40% |
Iron, Fe | 16.01 mg | 200.13% |
Magnesium, Mg | 294 mg | 70.00% |
Phosphorus, P | 810 mg | 115.71% |
Potassium, K | 2760 mg | 58.72% |
Sodium, Na | 17 mg | 1.13% |
Zinc, Zn | 4.05 mg | 36.82% |
Copper, Cu | 1.84 mg | 204.44% |
Manganese, Mn | 1.84 mg | 80.00% |
Selenium, Se | 5.9 µg | 10.73% |
Vitamins | Amount | % DV |
---|---|---|
Water-soluble Vitamins | ||
Vitamin B1 (Thiamin) | 1.656 mg | 138.00% |
Vitamin B2 (Riboflavin) | 0.355 mg | 27.31% |
Vitamin B3 (Niacin) | 3.597 mg | 22.48% |
Vitamin B5 (Pantothenic acid) | 1.654 mg | 33.08% |
Vitamin B6 (Pyridoxine) | 0.526 mg | 40.46% |
Vitamin B9 (Folate) | 817 µg | 204.25% |
Folate, food | 817 µg | N/D |
Folate, DEF | 817 µg | N/D |
Fat-soluble Vitamins | ||
Vitamin A, IU | 31 IU | N/D |
Vitamin E (alpha-tocopherol) | 0.39 mg | 2.60% |
Vitamin K (phylloquinone) | 10.3 µg | 8.58% |
Lipids | Amount | % DV |
---|---|---|
Fatty acids, total saturated | 0.427 g | N/D |
Myristic acid 14:00(Tetradecanoic acid) | 0.002 g | N/D |
Palmitic acid 16:00 (Hexadecanoic acid) | 0.401 g | N/D |
Stearic acid 18:00 (Octadecanoic acid) | 0.026 g | N/D |
Fatty acids, total monounsaturated | 0.144 g | N/D |
Oleic acid 18:1 (octadecenoic acid) | 0.144 g | N/D |
Fatty acids, total polyunsaturated | 0.712 g | N/D |
Linoleic acid 18:2 (octadecadienoic acid) | 0.388 g | N/D |
Linolenic acid 18:3 (Octadecatrienoic acid) | 0.324 g | N/D |
Amino acids | Amount | % DV |
---|---|---|
Tryptophan | 0.464 g | 105.45% |
Threonine | 1.645 g | 93.47% |
Isoleucine | 1.726 g | 103.23% |
Leucine | 3.122 g | 84.47% |
Lysine | 2.685 g | 80.29% |
Methionine | 0.589 g | N/D |
Cystine | 0.425 g | N/D |
Phenylalanine | 2.114 g | N/D |
Tyrosine | 1.1 g | N/D |
Valine | 2.046 g | 96.88% |
Arginine | 2.421 g | N/D |
Histidine | 1.089 g | 88.39% |
Alanine | 1.639 g | N/D |
Aspartic acid | 4.729 g | N/D |
Glutamic acid | 5.962 g | N/D |
Glycine | 1.527 g | N/D |
Proline | 1.658 g | N/D |
Serine | 2.127 g | N/D |
*Above mentioned Percent Daily Values (%DVs) are based on 2,000 calorie diet intake. Daily values (DVs) may be different depending upon your daily calorie needs. Mentioned values are recommended by the U.S. Department of Agriculture. They are not healthbenefitstimes.com recommendations. Calculations are based on average age of 19 to 50 years and weigh 194 lbs. Source: https://ndb.nal.usda.gov/
Health Benefits of Black beans
Black beans are a healthy addition to the diet. As it contains high content of protein, it is excellent for vegetarians and vegans. Black beans are loaded with fiber which supports digestion. It helps to reduce cholesterol and promote overall heart health. Due to its complex carb and protein, the body takes a longer time period for digestion than simple carbs. This makes one feel fuller for a longer time period.
- Prevention of cardiovascular problems
Black beans protect heart health by lowering inflammation. It contains high content of phytonutrients such as flavonoids known as petunidin, delphinidin, and malvidin. The studies have shown to control lipid fat metabolism and support the excretion of cholesterol. Black beans have a high content of soluble fiber which is a dietary fiber that is related to counteract heart disease by balancing the unhealthy levels of cholesterol. Studies have shown that a diet rich in dietary fiber from legumes and bean sources helps to protect against cardiac arrest, heart disease, and stroke. Fiber helps to lower inflammation as it eliminates toxins and waste from the body by clearing arteries from the buildup of harmful plaque. Studies have shown that one serving daily (3/4 cup of cooked beans) helps to lower the risk of heart attack and balances bad cholesterol. Moreover, Black beans are a great source of magnesium, folate which is vital for maintaining a healthy cardiovascular system. Black beans have a high content of fiber which prevents overconsumption as well as gaining excess weight around the heart.
- Source of antioxidants
Black beans have essential flavonoid and phytochemical compounds that act as anti-inflammatories and antioxidants which are helpful for protecting various forms of cancer especially colon cancer. Black beans having deep black color are the richest source of antioxidants. Antioxidants help to counteract damage made by free radicals or oxidative stress. When free radicals are excessively present in the body and standard western or poor diet does not provide adequate antioxidants for counteracting free radical damage then the disease is more likely to develop. Brightly colored vegetables and fruits have a high content of antioxidants which is also found in deeply colored black beans as well as other legumes. Studies show that high antioxidants found in Black beans assist in preventing DNA damage and gene mutation which could lead to lower chances of cancerous cells development.
- The high content of fiber
Beans are a great source of fiber. Black beans have a high content of dietary fiber which provides the feeling of satiety after consuming and assist in controlling blood sugar levels. Animal studies show that consumption of Black beans assists in controlling appetite and results to a significant reduction in unhealthy body fat. For maintaining good digestive health, adults should consume 30 grams of fiber every day. So consume one cup of black beans. The protein and fiber assist glucose from the starch of beans to release slowly into the bloodstream. Beans keep you full for a longer time period.
- Digestive health
The fiber content in Black beans promotes digestion by clearing the digestive tract from the buildup of toxins. It acts as a boon for the digestive system which pushes waste through the digestive tract so that the bacteria found in gut flora is balanced. It prevents unwanted health problems such as IBS and constipation. Studies have shown that Black beans protect colon health and prevent colon cancer due to their ability to obstruct oxidative damage in the digestive system. Fiber maintains the pH level of the body, balances alkalinity, and acidity. It has an alkalizing effect that balances pH levels by eliminating the high level of acidity. Studies have shown that Black beans clear unwanted bacteria and toxins from the body which restores digestive activity and enhances overall health. Fiber assists in nutrient absorption and release acids to the bloodstream, providing fuel, liver function, cleansing the digestive tract, and eliminating pathogens, harmful waste, extra sugar, and unhealthy cholesterol. Consume adequate fiber from plant sources for stimulating digestion, balancing electrolytes, and maintaining a healthy metabolism.
- Provide energy
Consume the right type of carbohydrate in form of whole and refined vegetables, legumes, starches, and sprouted grains for stabilizing blood sugar levels. Beans contain a complex carbohydrate known as starch, which the body digest slowly and use for energy without spiking level of blood sugars. It makes Black beans low on the glycemic index.
- Stabilize blood sugar levels
Black beans have starch which contains glucose, which is used by the body easily for various functions. Simple or fast carbs do the opposite of complex carbs such as starch-raising blood sugar levels quickly as the high sugar content is released at once to the blood. Intake of the wrong type of carbohydrate results in dips and spikes in energy levels which results in sudden sugar high after consumption of simple carbs which follows sugar crash. It leads to a low level of energy and puts chronic stress on the body. Black bean provides time-released energy in form of starches which makes it a great source of carbohydrate for people with resistance to insulin or diabetic patients.
- Great source of vitamins and minerals
Black beans are a great source of minerals and vitamins. It has a high content of iron, magnesium, and high content of phosphorus, folate, and Vitamin B. It is essential for vegans and vegetarians who may be lacking in these nutrients with the elimination of animal sources. Black beans have an anti-inflammatory activity that prevents fibromyalgia, metabolic syndrome, leaky gut, and others. Foods rich in magnesium are essential for maintaining cellular health as well as 300 biochemical functions in the body.
- Protein content
One cup of Black beans offers 14 grams of fat-burning protein with essential nutrients and antioxidants that help to slow down the aging process. The body uses protein in form of amino acids for various functions. Consume an adequate amount of protein daily to counteract symptoms associated with protein deficiency including fatigue, muscle weakness, eye problems such as cataracts, low energy, poor skin health, heart problems, and imbalanced hormones. Protein is essential for building muscles and makes the body energetic, youthful, and strong. Beans add high fiber protein and low fat to the diet.
- Lose weight
Black beans have a low amount of calories with adequate nutrients and fiber. Consume black beans to prevent overeating since fiber expands in the digestive tract, soaks water by taking high volume. It lowers food cravings for sweets or snacks and processed junk foods between meals. It is beneficial for those who want to lose weight and watch their intake of calories.
- Prevention of Birth defects
Pregnant women should consume a diet high in folic acids which assist in preventing birth defects. 400 micrograms of folic acid before or early into pregnancy lowers the chances of neural tube defects in infants which could be lowered by 70%. Neural tube defects such as encephalocele, anencephaly, and spina bifida could develop within the first 28 days of pregnancy. Add black beans to the diet as it offers daily recommended intake having just one cup. But remember not to consume more than 1000 mcg of folic acid per day as it could lead to deficiency in Vitamin B12.
Traditional uses
- Green pods are diuretics and help to lower the level of blood sugar.
- It is used for treating diabetes.
- Use the grounded flour externally for treating ulcers.
- Seeds are used for treating blood cancer.
- It is helpful for arthritis, rheumatism, and urinary tract disorders.
- It helps to prevent constipation and prevent digestive problems such and diverticulosis and irritable bowel syndrome.
- In traditional Chinese medicine, Black beans are used to lower knee pain, back pain, seminal emissions, infertility, ear problems, blurry visions, and promote diuresis.
Precautions
- Black beans contain purines so if used excessively, it could cause health problems. Purines break down to form uric acid, so excessive accumulation leads to gout as well as the formation of kidney stones.
- It contains polyphenols that might have negative aspects.
Side effects of Black beans
- Contains toxin
Black beans contain a toxin known as phytohemagglutinin. In comparison to red kidney beans, the amount of toxin is somewhat low and not considered to be dangerous. But it is recommended to consume Black beans after cooking instead of consuming raw as the cooking process lowers the toxicity level of this toxin.
- Contains Phytic Acid
Phytic acid does not cause direct harm to the body but its existence could interfere in the absorption of minerals which is present in Black beans. Remove phytic acid which acts as one thin layer of protection of seeds.
- Contains oligosaccharides
Oligosaccharides are complex sugar that could not be broken down easily in the system. It causes serious digestion problems. It is possible of one is consuming Black beans in high amounts. Nonetheless, the cooking process lowers the level of this substance.
How to Eat
- Beans are added to salads or prepared as vegetables.
- Soak the dried beans and add them to stews, soups and meat dishes.
- Powdered seeds are added to soups.
- Immature seeds are steamed or boiled and used as vegetables.
- Use the sprouted seeds in salads.
- Due to its dense and meaty texture, it is used in vegetarian dishes such as frijoles negros.
- In Cuba, it is a vital ingredient of Moros y Cristianos.
- It is widely used as an ingredient for soups.
- Black bean soup is served with white rice in Cuba.
- In some vegan brownie recipes, Black beans are used instead of eggs.
- Make bean soup by mixing cooked Black beans with onions, tomatoes, and spices.
- Layer black beans, guacamole, diced onions, chopped tomatoes, and cilantro in a serving bowl for making the delicious layered dip.
Black turtle beans (Side dish)
Ingredients:
- 2 lbs. black turtle beans
- 5 cloves garlic (chopped)
- 1 large onion (chopped)
- 4 cups vegetable broth (low sodium)
- 2 tsp chili powder
- 4 cups water
- 2 tsp ground cumin
- ½ tsp pepper
- ½ tsp salt
Instructions
- Remove debris if any from beans.
- Then soak black beans overnight for about 10 to 12 hours in six cups of water.
- Drain and rinse the beans.
- Add vegetable broth, black beans, water, onions, chili powder, garlic, pepper and ground cumin in a dutch oven.
- Heat over medium heat till boiling.
- Cover and lower the heat. Cook it over low heat for about 1½ hours.
- Add salt and continue to cook for more 30 minutes or till beans are tender.
Cuban-Style Black Beans
Ingredients
- 1 pound dried black beans (rinsed)
- 1 medium onion (finely chopped)
- 1 bay leaf
- 1 green bell pepper (seeds and ribs removed), finely chopped
- 8 garlic cloves (finely grated)
- 1 tablespoon of Morton kosher salt
- 1 teaspoon of dried Mexican or Italian oregano
- Freshly ground black pepper
- ¼ cup of extra-virgin olive oil
- Cilantro leaves with tender stems
Directions
- Boil five quarts of water with ¼ cup chopped onion, bay leaf, beans, 1 tbsp. grated garlic, ¼ cup of chopped bell pepper, ½ tsp. oregano, 1 tbsp. of Moron salt in a large pot.
- Cook by reducing heat to medium, stir occasionally and add more hot water if needed to cover till the beans are tender and cover by ¼ inch liquid for 2 ½ to 3 hours. Then discard bay leaf.
- At the same time, heat oil in a medium skillet over medium to low. Then add bell pepper, onion, oregano and garlic. Season with salt and pepper. Stir till the onions are soft and stir into cooked beans. Top by using cilantro.
How and Why to Cook Black Beans From Scratch
We go through a lot of black beans around here. We add them to vegetarian tacos, grain salads, and use them to make our favorite black bean soup, our quick and easy black bean burgers (vegan recipe), as well as these homemade veggie burgers. We love the ease of canned beans (and still use them), but my absolute favorite black beans are cooked from dried. Here are two reasons why:
- Just like with other dried bean recipes (like this dried chickpeas recipe), when you cook dried black beans from scratch, you have to opportunity to add flavor with spices, garlic, and other aromatics. This is why home-cooked beans taste so delicious!
- You are able to control how soft the beans are so if you want the beans to have some texture, which would be ideal for adding to soups or salsas (like this black bean and corn salsa), you can take them off of the heat before they get too soft. Alternatively, if you are turning the beans into a dip or spread, you can cook the beans a bit longer.
Cooking dried black beans from scratch is easy! Here’s our go-to three-step process for doing it on the stove (my favorite method):
- Add beans to a pot with aromatics (like garlic, onion, bay leaves, and spices) and enough water to cover the beans by 3 to 4 inches.
- Slowly simmer the beans until tender, which takes 1 1/2 to 2 hours.
- Remove aromatics simmer a bit longer to make a creamy sauce, and then enjoy!
How to Use Home Cooked Black Beans
Our recipe is for one pound of black beans (about 2 cups dried). One pound of dried beans makes 4 1/2 to 5 cups of cooked black beans. The beans will keep, stored in an airtight container in the fridge for 3 to 5 days and in the freezer for a few months.
I love to spoon these creamy cooked black beans over rice and grains (see our cilantro lime rice or this tutorial for cooking quinoa). The slow simmer creates a thick creamy sauce, which thanks to the aromatics tastes amazing.
For some dishes, though, you might find that the sauce isn’t needed. Take our homemade black bean burgers, for example. In that recipe, these flavorful beans would be perfect, but too much sauce would make the burgers mushy. So when following that recipe, I drain the beans first and rinse them before adding them to the burger mixture. You can always save the sauce for another recipe.
More Ways To Cook Black Beans
I’ve already mentioned that cooking black beans on the stovetop is my preferred method. I wish it was a little quicker, but it’s the best method I’ve found that creates that thick, flavorful, and creamy sauce.
That said, when I’m short on time, I do love using my pressure cooker to cook beans (we have a 6-quart Instant Pot). We’ve shown us using it to make refried beans and to cook chickpeas ready for hummus. I’ve shared the steps for cooking black beans in a pressure cooker in the recipe below.
More Easy Bean Recipes
- Easy Creamy Coconut Black Beans — Thanks to one pot and coconut milk, these easy creamy coconut black beans are made completely from scratch.
- Black Bean and Quinoa Salad — Thanks to lots of vegetables and quinoa, this easy black bean salad is ultra satisfying, hearty, and absolutely delicious.
- Utterly Delicious Chipotle Bean Chili — Make this easy and delicious vegetarian bean chili with dried beans or canned.
- Kale and Bean Salad with Tahini Dressing — This easy kale and bean salad is packed with good for you ingredients, is crave-worthy and can be made in advance.
- Refried Beans (Use Pinto or Black Beans) — How to make delicious homemade refried beans from scratch. We’re showing you how to make refried beans using dried pinto or black beans as well as how to make refried beans from canned beans.
How to Cook Black Beans From Scratch
- PREP
- COOK
- TOTAL
Our favorite black beans recipe that can be made in one pot on the stove. These beans are ultra-flavorful, creamy, and taste so much better than canned. Tips for adapting our recipe for a pressure cooker is provided in the recipe below.
How to know when the beans are done: How soft the beans should be depends on what you plan to do with them. For dips and spreads, cook the beans until they are very soft and you can crush them easily with your fingers or a fork. For salads, topping rice and grain bowls, and for soups, cook the beans until tender, but not falling apart.
YOU WILL NEED
- 1 pound (450 grams) dried black beans, rinsed (about 2 cups)
- 3 medium cloves garlic, peeled and gently smashed, use more for garlicky beans
- 1/2 medium peeled onion, see notes
- 1 bay leaf
- 1 medium lime, halved
- Water
- Salt and fresh ground black pepper
DIRECTIONS
STOVETOP METHOD
Add the beans, garlic cloves, onion half, and bay leaf to a large pot and cover with 3 to 4 inches of water. Cover and bring to a boil over high heat. Uncover, stir the beans, and reduce the heat to keep the beans at a low simmer.
Cook, stirring occasionally, at a low simmer until the beans are tender, 1 1/2 to 2 hours. As they cook, if you notice the beans look a little dry, add a bit more water to the pot.
Remove the garlic cloves, onion, and bay leaf. Add the juice of half a lime and season the beans with salt. Start with 1/4 teaspoon and go from there based on your tastes.
Serve straightaway or to make the creamy gravy that surrounds the beans in our photos, increase the heat to medium, and then simmer until the liquid around the beans has thickened and sticks to the beans. Serve with additional lime juice squeezed over the top if needed.
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- PRESSURE COOKER METHOD
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Rinse the dried beans then add to a 6-quart pressure cooker (we have an Instant Pot). Add the garlic cloves, onion half, and bay leaf. Add enough water so that the beans are covered by 1 1/2 inches, but do not fill past the “max fill line.” Add 1 teaspoon of neutral oil like vegetable or olive oil (this prevents the beans from foaming).
Seal the lid and cook on High Pressure for 30 minutes, and then allow the cooker to naturally release (you can tell when it is safe to open the cooker when the steam release valve drops). We recommend you read the user manual of your specific pressure cooker before using this method.
Remove the onion, garlic cloves, and bay leaf, and then discard them. Add the juice of half a lime and season the beans and liquid with salt. Start with 1/4 teaspoon and go from there based on your tastes.
Serve straight away or to make the creamy gravy that surrounds the beans in our photos, turn the pressure cooker to SAUTE and simmer, without the lid until the liquid thickens and starts to stick to the beans. Serve with additional lime juice as needed.
Best Tips for Cooking Dry Black Beans
- Go ahead and add salt – If you’re familiar with cooking dry beans, you might be surprised that I add the salt right away. Rumor has it that salting the beans too soon can lengthen the cooking process, but in my experience, it doesn’t make a difference. I’ve even cooked two pots of dried beans side by side, salting one but not the other. The timing was exactly the same, but the flavor in the salted pot was way better. Don’t be scared to salt!
- Fresher beans = better beans – The primary determinant of how long it takes to cook beans from scratch is NOT when you add the salt: it’s the freshness of the beans! If you can, source them from a grower or store with high turnover. The fresher your beans are, the better they’ll be.
- Add kombu for more digestible beans – Simmering dried beans with a piece of kombu kelp is said to make them more digestible, so if you like, feel free to add some after the pot comes to a boil and you reduce the heat to a simmer. Be careful not to boil the kombu – if you do, it can give the broth a bitter flavor.
- Save some for later – If you’re cooking black beans from scratch, you might as well make a big batch! Eat some right away, and save the leftovers in an airtight container in the fridge for up to 5 days. You can also freeze them for up to 3 months. Feel free to drain them or store them in their cooking liquid. Both ways work!
- Onion – So that the onions are easy to remove after cooking, slice the onion through the root and leave enough of the root intact so that the layers of onion do not fall apart.
- Optional Soaking – Before cooking the beans, you can soak them overnight. We do not find this is necessary, but some may find that an overnight soak helps with digestibility and more even cooking. To do it, add beans to a big bowl with enough water that it covers them by 3 to 4 inches, and then set aside to soak overnight. The beans will triple in size.
- This recipe was inspired by Serious Eats.
- Nutrition facts: The nutrition facts provided below are estimates. We have used the USDA database to calculate approximate values.
References