Step 1Place the lamb, water, carrots, celery, yellow onion, peppercorns, bay leaves, salt and pepper in a 6-quart saucepan and bring to boil. Lower the heat, cover and simmer until the meat is tender, 1 1/2 hours.
Step 2Remove the meat; strain the juices into a bowl. Discard the vegetables. Put the bowl in the freezer until the fat has solidified enough to skim from the top, about 45 minutes. Discard the fat and cut the meat from the bones. Chop the meat into small pieces; set aside.
Step 3Pour the defatted juices into a small, 4-quart saucepan and bring to a boil. Add the bouillon cubes and stir to dissolve. Remove from the heat and set aside.
Step 4Melt the butter in a 6-quart saucepan over low heat. Add the green onions and cook until tender, about 4 minutes. Stir in the flour until blended. Add the dill, parsley and rice and mix well. Cook on low heat for a couple of minutes and add the chopped meat. Pour in the broth. Cover and bring to a boil, then lower the heat and simmer until the rice is done, 15 to 20 minutes.
Step 5Break the eggs into a small bowl and beat with a fork. Gradually beat in the lemon juice. Add a ladle of the soup from the pot to the egg mixture while beating the eggs. Gradually add a second ladle. Pour the egg mixture into the soup gradually while continuously stirring. Cook the soup another 3 minutes; do not allow it to boil. Remove from the heat.