Combine sausage and chopped onion in a medium skillet; cook over medium heat until browned, stirring to crumble meat. Drain. Combine sausage mixture, stuffing mix, chopped celery leaves, parsley, 2 tablespoons water, salt, and pepper; mix well. Set aside.
Flatten cutlets to 1/8-inch thickness, using a meat mallet or a rolling pin; place 1/4 cup stuffing mixture on each veal cutlet. Fold cutlet over stuffing mixture, and secure with wooden picks.
Dredge each veal bird in flour, and brown in butter in a large skillet. Add remaining water, sliced onion, and mushrooms; cover and simmer 45 minutes or until veal is tender. Transfer veal birds, onions, and mushrooms to a warm serving platter; garnish with celery leaves, if desired.