To prepare the ragu di salsicce, in a medium pot over medium-high heat, melt the butter and warm the olive oil. Add the onions and cook, stirring occasionally, until translucent. Add the celery and carrots and cook, stirring occasionally, for 3 minutes.

Add the sausage and cook, stirring occasionally, until the meat is lightly browned. Add the milk and nutmeg and cook, stirring occasionally, for 5 minutes. Stir in the tomatoes with juices, reduce the heat, cover partially with the lid and simmer for 2 1/2 to 3 hours.

Remove from the heat and adjust the seasoning with salt and pepper.

To make the béchamel sauce, in a saucepan over low heat, melt the butter. Whisk in the flour and cook gently for 8 to 10 minutes.

Remove the pan from the heat and vigorously whisk in the milk. Return the pan to medium-high heat and cook, stirring constantly, until the sauce is thick.

To assemble the lasagna, grease a baking pan with butter. Spread a layer of the lasagna noodles on the bottom of the pan, then spread a layer of the ragu and sprinkle with a little Parmesan. Add another layer of noodles with a layer of the béchamel on top. Three layers of ragu and two layers of béchamel. Top with a layer of ragu, then sprinkle with the remaining mozzarella and Parmesan and the Fontina.