Preheat oven to 350 degrees F (175 degrees C). Spray a 13x9-inch baking dish with non-stick cooking spray.
Heat butter in a skillet over medium heat. Stir in mushrooms, onion, and pepper. Cook until soft and tender. Sprinkle flour over vegetables, and stir to coat. Stir in milk and chicken broth. Cook until sauce thickens. Stir in broccoli, Parmesan, salt, pepper, and nutmeg. Cook until Parmesan is melted.
Layer baking dish with 1/4 sauce, 1/4 noodles, 1/4 ham, 1/4 chicken, and 1/4 Swiss. Repeat layers 3 times. Cover with foil.