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If beets are large, cut them in half. Put the beets into a large stockpot and cover them with water. Bring water to a boil, and simmer until the beets can be pierced with a butter knife (anywhere between 25 and 45 minutes). Drain the beets, and under cold running water, squeeze off the skins and cut beets into quarters.
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Place beets in a food processor or blender. Toast the coriander and cumin in a small dry skillet, shaking it often until they release their fragrance. Add the spices and peppers to the beets. Add 1/2 cup of the water, and blend the mixture until the large chunks are gone, but not to a smooth puree (you want some texture to this soup). Transfer the beet mixture to a large bowl.
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In the processor or blender, combine the cucumber, onion, and tomatoes. Add 1/2 cup of the water. Process until the mixture is only slightly chunky. Add this mixture to the beet mixture. Add the remaining 1 cup water, the lime juice, the olive oil, the cilantro or basil, and the salt and pepper. Chill the soup for at least 1 hour. Then ladle it into bowls, garnish each bowl with a tablespoon of yogurt or sour cream, and serve.