Preheat oven to 350 degrees. Place chicken between plastic wrap and flatten to 1/4 inch thickness. Mix together flour, salt, and pepper. Dredge chicken lightly in flour mixture. In a large skillet over medium-low heat, melt 3 tablespoons butter. Place several pieces of chicken in the pan, without overcrowding them. Cook for 3 to 4 minutes on each side, or until golden brown. When browned, place chicken in a 13-by-9-by-2-inch pan. Repeat browning procedure with remaining chicken, adding 2 tablespoons butter each time. Reserve drippings.
Saute mushrooms in the same pan with 2 additional tablespoons butter. Stir wine and chicken stock into the pan, scraping to deglaze the pan, and cook for about 10 minutes until sauce is reduced by a third. Spoon sauce over chicken. Combine Monterey Jack and Parmesan cheeses and sprinkle over chicken.
Bake for 20 minutes or until thermometer reaches 165 degrees. Place under broiler until cheese has melted and is slightly browned. Depending on the size of the pan, you might need to brown the chicken in several batches. This will affect the amount of butter needed and the cooking time.