In a pot, melt the butter and add the flour. Mix until you get a paste and cook it for a minute or two until it gets thicker.
Add the room temperature or warm milk and turn down the heat. Mix occasionally, to melt the roux and to make sure it doesn’t stick to the pot.
Let it simmer for 15-20 minutes until it gets thick enough to coat a spoon, then season with salt, black pepper and a pinch of ground nutmeg to taste.
Assemble the layers:
Preheat the oven at 200°C (400°F).
Spread a thin layer of white sauce on the bottom of the pan, cover with a layer of pasta, then a layer of Ragù sauce, a layer of bechamel sauce and grated Parmigiano.
Repeat the layers, pressing down a little bit when you add the pasta. When the pan is full, finish with a last layer of pasta, sauce and a generous amount of grated Parmigiano cheese.
Bake in the middle rack of the oven for about 35-40 minutes. If the crust is not crunchy enough, finish with 3-4 minutes under the broiler. If instead you notice the crust getting too brown before the cooking time is over, cover the pan loosely with foil.