Wash and pat dry the chicken breasts. Put each chicken breast between two layers of plastic wrap, on a chopping board and pound them to a thickness of approximately 1/4 inch.
Lay the chicken breasts flat; season each side with salt and black pepper. Sprinkle the mozzarella cheese over each breast.
Place the slices of ham on top, covering the surface of the breasts.
Meanwhile, microwave cream cheese for 30 seconds. Add chopped garlic and mix well. Spread the cream cheese mixture over the ham.
Roll tightly, securing with toothpicks if needed.
Preheat oven to 350 degrees F. Put flour, beaten egg and bread crumbs on to 3 separate flat dishes. Dredge each chicken breast roll first in the flour then the beaten egg and finally the bread crumbs.
Transfer chicken breast rolls to a greased baking dish, seam side down. Set a tablespoon of butter on top of each and bake at 350 degrees F for 50-60 minutes, depending on the size of the breast, until the juices run clear, and the chicken is slightly browned. Serve with Mushroom Sauce.