Put a thin layer of bechamel in the bottom of a shallow baking dish or a jelly-roll pan. Cover the bechamel with a thin layer of ragu and a little cheese. Top the sauces and cheese with a layer of lasagna noodles. Repeat 6 times ending with a layer of bechamel, ragu and cheese. Bake the lasagne until it is warm at the center and the cheese topping golden brown, about 45 minutes (cover with some aluminum foil if the lasagna browns before it is fully heated).
Whisk the flour mixture into the milk. Whisk constantly as the sauce returns to a boil and simmer until the sauce thickens sufficiently to coat the back of a spoon. Season the bechamel with salt and reserve it in a water bath until ready to use.
Divide the dough in thirds. Roll each sheet of dough through the machine starting at the widest setting then graduating to thinner settings and stopping when the sheets of dough are a medium thickness. Place the sheets of pasta on a lightly floured surface and set aside until ready to use.