Flatten chicken to 1/4 inch thickness. Place egg in a shallow bowl. Combine cracker crumbs and salt in another shallow bowl. Dip chicken in egg, then in crumbs, set aside.
In a large ovenproof skillet, saute mushrooms in 1 tablespoon butter until tender; remove and set aside. In the same skillet, melt remaining butter. Add chicken; cook over medium heat for 3-4 minutes each side or until juices run clear. Top each chicken breast half with a ham slice, mushrooms and a cheese slice. Broil 4 inches from heat for 1-2 minutes or until cheese is melted.
If you do not have an ovenproof skillet, just remove the chicken from the skillet and put in a 9x13 inch glass dish sprayed with cooking spray before you place them in the oven.