Heat the oven to 375 F. Boil the noodles until just tender, then drain and lay them flat so they do not stick.
Stir the ricotta and egg together in a bowl. Spread a thin layer of ragù in a 9-by-13-inch baking dish.
Layer noodles, ragù, béchamel, ricotta, Parmesan, and mozzarella. Repeat for three layers, finishing with béchamel, Parmesan, and mozzarella.
Cover with foil and bake for 30 minutes. Uncover and bake for 15 to 20 minutes more, until bubbling and browned.
Rest for at least 15 minutes before slicing and serving.