Whisk bread crumbs, 3 tablespoons Parmesan cheese, and oregano together in a shallow dish until evenly mixed.
Dip each chicken breast into ranch dressing. Transfer to the bread crumbs and coat evenly.
Place chicken breasts in a baking dish. Sprinkle Parmesan cheese on 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, 15 to 20 minutes. Remove aluminum foil from the baking dish. Bake until cheese starts to brown, 5 to 7 minutes more.