Heat 2 tablespoons butter in large skillet over medium heat. Add mushrooms and onion; cook 7 minutes or until liquid from mushrooms evaporates, stirring frequently. Stir in wine, 1/4 teaspoon salt and 1/4 teaspoon pepper; continue cooking 5 minutes or until wine evaporates. Remove from heat; cool.
Sprinkle remaining salt and pepper over chicken. Place one slice prosciutto over each piece of chicken.
Dip both sides of chicken in flour to coat lightly. Melt remaining butter with oil in large skillet over medium heat. Place chicken, prosciutto-side down, in skillet; cook 3 minutes. Turn; continue cooking 3 more minutes or until chicken is browned.
Transfer chicken to 13x9-inch casserole dish coated with cooking spray. Spread mushroom mixture evenly over chicken. Drizzle cream over mushroom mixture, if desired. Place one slice cheese over each piece of chicken. Bake in preheated 400°F oven 10 minutes or until chicken is cooked through and cheese is melted.