COMBINE CHICKENS AND WATER in a 5-quart stockpot. Cover, bring to a boil over high heat. Reduce heat to low and skim off the scum that accumulates on the surface. Remove chicken breast quarters and set aside. Add 2 cups carrots, 1 cup onions, 2 cups celery, green leek tops, peppercorns, bay leaves, thyme and salt. Cover and simmer for 45 minutes. Remove cover, replace the chicken breasts and continue to cook for another 25 minutes. Meanwhile, slice the white parts of leeks into 1/4-inch rounds and wash well to remove any sand. You should have about 1 cup. Melt the butter in a medium saucepan over medium heat and add the leeks and the remaining carrots, onions and celery. Cover and cook gently until vegetables are soft, about 5 minutes.
Remove the chicken from the liquid and place on a platter. Strain the liquid through a fine strainer into a large plastic container and discard the herbs, spices and the cooked soup vegetables. Skim and discard any fat from the surface of the broth. Remove and discard chicken skin. Remove the meat from the bones, dice it and set aside. Place bones in an airtight freezer container and save for making stock. Immediately prior to serving, transfer broth to a pot and heat, covered, over high heat. Heat chicken and vegetables, covered, in 350F oven until hot. Decorate each soup bowl with some heated diced vegetables and chicken. Pour the hot broth into a pitcher or soup tureen and pour the soup into the garnished bowls at the table.