In a 13 x 9 x 2-inch baking dish, spread 1/2 cup of the béchamel sauce on the bottom. (Set aside 1 cup of the béchamel sauce for the top.) Arrange a single layer of lasagna noodles over sauce; spread some of the filling over noodles, top with a sprinkling of the grated cheeses. (Make sure to reserve a bit of cheese for the topping.) Repeat layers, ending with noodles.