Place chicken breasts in a single layer in the center of a 9x13-inch baking dish. Pour green beans into one side of the baking dish. Spread potatoes out on the other side of the dish.
Sprinkle dressing mix over chicken, beans, and potatoes; drizzle butter over the top. Cover baking dish with aluminum foil.
Bake in the preheated oven until chicken breasts are no longer pink in the center and the juices run clear, about 1 hour. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).