Flatten chicken to 1/4-in. thickness. In a shallow bowl, lightly beat the egg. Combine cracker crumbs and salt and pepper in another shallow bowl. Dip chicken in egg, then roll in crumbs; set aside.
In a large ovenproof skillet, saute onions and mushrooms in 1 tablespoon butter until tender; remove and set aside. In the same skillet, cook chicken over medium heat in remaining butter for 3-4 minutes on each side or until no longer pink.
Top each chicken breast half with a ham slice, onions, mushrooms and a cheese slice. Broil 4-6 inches from the heat for 1-2 minutes or until cheese is melted.