Sauté veal, mushrooms, and onion in 1 tablespoon shortening in a large skillet until veal is no longer pink and vegetables are tender. Stir in flour.
Combine eggs, milk, parsley, salt, and pepper in a medium mixing bowl; beat well. Stir in veal mixture.
Melt remaining shortening in a 10-inch skillet. Pour egg mixture into skillet. Cover and cook over low heat 25 minutes or until set. Sprinkle with paprika, if desired.
Sauté sliced mushrooms in butter until tender. Arrange over omelet. Cut omelet into wedges, and serve immediately.