Preheat oven to 350 degrees F (175 degrees C). Grease a 13x9-inch baking dish.
Arrange chicken breasts into prepared baking dish. Top each breast with two slices Swiss cheese.
Stir chicken soup and white wine together in a bowl until smooth; spoon over chicken and cheese portions. Sprinkle crushed stuffing mix over the soup mixture. Drizzle melted butter over the stuffing.
Cook in preheated oven until no longer pink in the center and the juices run clear, about 55 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).