Preheat the oven to 175°C.
Butter a 13 by 9 by 4 baking dish and add a thin layer of white sauce. Cover with a layer of lasagna sheets, and then another thin layer of white sauce. Gently spread about 4 tablespoons pesto across the surface, and then top with about 2 tablespoons Parmesan. Repeat until you finish layering the lasagne. Top with a final layer of lasagne sheest and spread a final very thin layer white sauce, pesto, Parmesan cheese and a little butter to help the cheese crisp-up when cooked in the oven.
Cook the lasagne for about 30 minutes. Serve dressed with some Parmesan and a drop of extra-virgin olive oil.
Combine the basil leaves, pine nuts, garlic, and olive oil in a mortar and pestle and pound until paste forms. Add the Parmesan cheese, Pecorino cheese, salt and pepper and stir until smooth.
Keep the pesto in the refrigerator until you're ready to use it.
For the white sauce:
Melt the 1/2 cup butter in a pan over medium heat. Stir in the flour with a wooden spoon. Cook's Note: This is an important moment, as you have to slowly toast the flour without burning it. This will help you lose the floury taste.
Warm up the milk and gradually ladle into the pot with the butter-flour mixture, whisking constantly while bringing the mixture to a boil. Reduce the heat, and simmer for about 15 minutes. Season the sauce with freshly grated nutmeg, salt, and pepper.
This is a crucial moment for your pesto, as you need to season with salt and pepper. Keep in mind that Parmigiano and Pecorino are both dry and salty cheeses. You have to taste your sauce a few times in order not to go overboard when you are seasoning it. If the sauce is too thick, add a little more milk, if too runny, return to the heat and add a pat of butter mixed with an equal amount of plain flour. The most important thing though is: white sauce should not taste floury. If you think your sauce is ready, but you can spot a hint of "flouriness" when you taste it, think again, and keep on cooking it for a few minutes more.