1 pound lasagne noodles, cooked until very al dente and drained, or no-boil lasagne noodles, uncooked
Unsalted butter, for greasing the dish
1 cups finely grated Parmesan
Make the béchamel sauce:
Melt the butter in a small saucepan over medium heat. Right when the butter begins to foam, add the flour and cook, whisking until the roux turns light golden brown and starts to smell nutty, 4 to 6 minutes. Pour in the milk, 1 cup at a time, and bring to boil. Reduce the heat and simmer, whisking often so that the bottom doesn’t scorch, until the sauce thickens, 12 to 15 minutes. Turn off the heat and season with salt, white pepper and nutmeg.
Make the ragù bolognese:
In a food processor, combine the celery, carrots and onion and pulse until finely chopped.
In a large, heavy-bottomed pot over medium heat, add 2 tablespoons of the olive oil. When the oil is hot, add the pancetta and cook, stirring until some of the fat has rendered, about 5 minutes. Add the chopped vegetables and season with salt. Cook, stirring constantly until the vegetables are very soft and all the liquid has evaporated, 15 to 20 minutes. Transfer mixture to a bowl and set aside.
Return the pot to the stove over medium-high heat and add the remaining tablespoon of olive oil. Using a wooden spoon, break the sausage into small clumps. Season it with salt and brown, 15 to 20 minutes.
Return the reserved vegetable mixture to the pot with the bay leaf and thyme and cook, stirring often, until most of the liquid has evaporated, 10 minutes. Deglaze the pan by adding the wine, making sure to scrape the brown bits on the bottom of the pan; reduce until all the liquid has evaporated and the contents have taken on a rich chestnut color, 15 to 20 minutes. Add the tomato paste and cook for 3 to 4 minutes.
Deglaze the pan with 1½ cups of the stock and ½ cup of the milk and reduce by half, about 15 minutes. Add the remaining stock and milk, skimming any fat that rises to the surface, and continue cooking until the flavors have melded together and the sauce has thickened, 30 to 40 minutes. Season with salt and allow to cool. Remove and discard the bay leaf and thyme. Make ahead: The sauce can be made up to 3 days ahead. Allowing the sauce to cool completely then slowly reheating until just warmed through will give the sauce a deep, richer flavor.
Make the lasagne:
Preheat the oven to 350°. Reheat the sauces to warm if they were made ahead. Grease a 9- x 13-inch baking dish. Spread ½ cup of the béchamel sauce in the bottom of prepared dish; top with ¼ of the noodles. Top the noodles with ¾ cup bolognese sauce, followed by ¼ cup of the béchamel and ? cup of the cheese. Repeat the process 3 times with remaining noodles, sauces and cheese. Reserve any extra meat sauce for another use.
Transfer the baking dish to a sheet tray and bake until the cheese becomes brown and bubbly, 40 to 45 minutes. Remove from the oven and allow the lasagne to cool for 30 minutes before serving.