In a large bowl, combine the meat, grated onion, garlic, rice, 3 tablespoons chopped parsley, the mint, oregano, salt and pepper, and 1 egg, slightly beaten. Knead for a few minutes, then shape into walnutsized barrels and set aside. In a soup pot, bring the water or stock to boil with the chopped onion, celery and carrot, and salt and pepper to taste. Lower the heat and add the meat-rice barrels. Simmer, covered, for 30 minutes, then remove from the heat. To prepare avgolemono, beat the two remaining eggs for 2 minutes. Continuing to beat, gradually add the lemon juice. Then add 1 cup of the hot broth by droplets, beating steadily, until all has been added. Add to the soup and heat, being careful not to let it boil. Serve hot, garnished with parsley. Note: This soup is frequently made without the additional vegetables added to the liquid. Also, you may enjoy this soup without avgolemono, in which case add 1/2 cup canned tomato sauce to the liquid and reduce the water to 4 1/2 cups.