1. Place the dried mushrooms in a small bowl, pour the boiling water over them, and soak for at least 2 hours, or until the mushrooms are soft and flexible. Drain the mushrooms (save the water) and chop them coarsely.
2. In a heavy 3- to 4-quart casserole, combine the mushrooms and their soaking water, the chicken gizzards or hearts, the carrot, string beans and peas. Pour in the 2 quarts of cold water and bring to a boil over high heat.
3. Reduce the heat to low and simmer partially covered for about 20 minutes, or until the chicken pieces are done and the vegetables are tender but still intact.
4. Pour the contents of the casserole into a fine sieve set over a deep bowl, and set the chicken pieces and vegetables aside. Return the stock to the casserole.
5. Meanwhile, melt the teaspoon of butter over moderate heat. When the foam begins to subside, add the pearl barley and stir for 1 to 2 minutes, until the pearls glisten with butter.
6. Do not let the barley brown. Add the barley to the stock in the casseroleand bring to a boil over high heat.
7. Reduce the heat to low and simmer partially covered for 10 minutes. Add the potatoes and simmer partially covered for 20 minutes longer, or until the barley and potatoes are thoroughly cooked.
8. Return the chicken pieces and vegetables to the soup. Taste and season with salt and freshly ground pepper, then simmer for a few minutes to heat the chicken and vegetables through.
9. To serve, ladle the soup into a heated tureen or individual soup plates. Spoon the sour cream into a separate bowl or sauceboat, sprinkle the top with the dill, and serve with the soup.
