Preheat oven to 400 degrees F (200 degrees C). Lightly grease a large, shallow baking dish.
Arrange flounder fillets on prepared baking dish and brush with 1 tablespoon butter. Combine remaining 3 tablespoons butter with bread crumbs, Parmesan cheese, salt, pepper, and thyme in a bowl; sprinkle over flounder.
Bake in preheated oven until fish flakes easily with a fork, 15 to 20 minutes. Carefully transfer fillets to a plate with a spatula.