All sorts of seeds can be sprouted for culinary use; alfalfa sprouts, snow pea (mangetout) sprouts, mustard sprouts and the like, but when referring to Asian bean sprouts, only two varieties are commonly used. Mung bean sprouts and soybean sprouts.
Mung bean sprouts. The sprouted seeds of a small plant (Vigna radiata) that are also referred to as moong beans. Mung beans are quite small, only about 5 mm in diameter and have a green olive coloured seed coat. The sprouts reach a length of 5 - 8 cm and are a pale white to cream colour. These sprouts are best eaten raw or cooked very briefly to retain their crunch.
Soybean spouts. The multi-faceted soybean (Glycine max) is also used for sprouting. Compared to mung bean sprouts, they are quite large, about 10 - 12 cm in length and have a yellowish colour. The soybean attached at the end is about the same size as a peanut. These sprouts are almost always cooked, even for raw dishes. This is not only because of their relatively denser texture, but also due to a trypsin inhibitor, which impedes the action of trypsin, a pancreatic enzyme that aids in the digestion of protein. Some sources suggest blanching soybean sprouts for at least 4 minutes before eating to remove this inhibitor. The soy bean sprout has the highest protein to calorie ratio of any vegetable.
What all bean sprouts have in common is their fantastic crunchy texture. They are not so much used as a source of flavour, but for the textural interest they can bring to noodle, soup and stir fried dishes.
The end of all bean sprouts have a small brown stingy protrusion, which many sources say must be removed before cooking. This string has a habit of getting between the teeth of the diner, but it is a very labour intensive procedure to pinch them off. I will leave it to you whether to do this or not. I personally don't. Some up market restaurants actually go to the degree of pinching off the bean as well, leaving a slender, naked sprouted stem. These finicky things are called silver sprouts.
As Teiresias points out, sprouts have a high concentration of ascorbic acid, or vitamin C. For reasons that remain largely unknown, prodigious amounts of ascorbic acid are produced in actively growing plant tissue, like seedlings. The amount can be as 3 - 5 times the concentration of the seed itself, and that is including the effect of water dilution in the sprout.
How to sprout your own beans
Soak 3 Tablespoons of dried mung or soybeans overnight in cold water. Drain and place in a large glass jar, around 1 litre (4 cups) in capacity. Cover the jar with a piece of muslin or cheesecloth and secure with a rubber band. Place in a warm, dark spot. Twice a day, morning and evening, fill the jar with blood temperature water, then drain the liquid away. The sprouts will be ready in 5 - 7 days.
As an interesting aside, The eminent gastronome, Georges Auguste Escoffier stated in 1903 that;
"a prolonged soaking of dried vegetables may give rise to incipient germination, and this, by impairing the principles of the vegetables, depreciates the value of the food, and may even cause some harm to the diner"
Hmm..
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