Wash the meat and bones, and place into a heavy stockpot with the bunch of parsley, carrots, celery, and cold water. Bring to a boil and simmer gently, covered, for 2 hours. Remove from heat and strain well. (Feed the meat to the dog, or chop up for hash.) Return stock to the pot and boil smartly to reduce it to 1 quart. Skim off any fat. Add the diced vegetables and cook until tender, about 20 minutes. Add salt and pepper to taste.
Place butter, salt, and water into a heavy 1-quart saucepan and bring to a boil. Add flour and stir with a wooden spoon until dough forms a clump. Remove from heat, cool for about 5 minutes, and beat in whole egg. Beat in yolk and chopped parsley. Wet hands and form dough into small dumplings. Add the dumplings to the simmering soup and cook for about 5 minutes, uncovered.