Sprinkle both sides of the chicken with salt and pepper. Spoon flour onto shallow dish or pie plate. Coat both sides of chicken with flour; shake off excess.
In 12-inch skillet, heat oil over medium heat. Brown chicken on both sides. (You don’t need to cook the chicken completely because it will finish cooking in the oven.) Transfer chicken to 13x9-inch (3-quart) glass baking dish.
Place butter in same skillet; let melt over medium-high heat. Add mushrooms; sprinkle with salt and pepper. Cook and stir mushrooms until golden brown. Stir in sherry; cook and stir for additional 1 to 2 minutes. Stir in soup; mix well. Cook until thoroughly heated.
Pour mushroom sauce over chicken; top each piece of chicken with one slice cheese. Fold cheese in half if it is too wide for chicken piece.
Cover dish with foil; bake 30 minutes. After 30 minutes, turn oven to boil setting. Remove foil from dish; broil 1 to 2 minutes to brown the cheese. Remove from the oven; set aside to cool slightly.