Step 1In a medium, heavy-bottom saucepan, heat the butter over medium heat until it foams and subsides. Whisk in the flour and stir it well using a wooden spoon, incorporating it into the butter until a dry paste forms (a roux). Reduce the heat and continue to cook and stir, taking care not to let the mixture brown.
Step 2Meanwhile, in a large saucepan, bring the milk to a simmer, then gradually whisk it into the pan with the roux, continuing to whisk until the mixture is smooth. Season with 1/2 teaspoon salt and a pinch of pepper, or to taste, along with the nutmeg, if using. Continue to cook over medium-low heat, stirring regularly, until the sauce is thick enough to coat the back of a wooden spoon, about 10 minutes. This makes about 4 cups bechamel.
Step 1To make the Bolognese sauce, in a medium, heavy-bottom pot, heat 2 tablespoons oil over medium-high heat. Add the onion, carrots, celery, garlic and thyme and season with ½ teaspoon salt and several grinds of pepper, or to taste. Cook, stirring regularly using a wooden spoon, until the vegetables are tender and have released their juices, 7 to 9 minutes. Stir in the livers and cook over high heat for 2 minutes, then add the beef, veal and pork, stirring and breaking up over high heat. Season with 1 teaspoon salt and several grinds pepper, or to taste. Continue to cook over high heat until the meat is brown, stirring regularly and scraping the bottom of the pan as necessary to keep the meat and vegetables from scorching.
Step 2Once the meat is browned, stir in the tomato paste over medium heat. Cook for about 20 minutes, stirring regularly, to marry the flavors. Add the wine, bring to a boil and cook until the wine is reduced by half, then add the milk and bay leaves and bring to a boil. Reduce the heat to a simmer and cook for 1½ to 2 hours, stirring occasionally. You may need to add a bit of water (or chicken or veal stock, if you have it) to thin the sauce if it thickens too much.
Step 3Taste the sauce and season with 1 teaspoon salt and several grinds of pepper, or as needed. Remove from heat and stir to release the steam and allow it to cool slightly. Skim the fat off the top with a ladle and discard.
Step 4Heat the oven to 350 degrees.
Step 5Coat the inside of a 13-inch by 9-inch (or similar size) baking dish with the remaining tablespoon oil. Cover the bottom of the dish with a layer of bechamel. Sprinkle over some grated cheese, then top with a layer of noodles. top the noodles with a layer of Bolognese sauce, and repeat with the bechamel, grated cheese, noodles, and Bolognese until the pan is filled to the top. The top layer should be Bolognese, dotted with bechamel, with thin slices of mozzarella laid across the top.
Step 6Place the baking dish on a foil-lined sheet pan and bake in the oven until the lasagna is browned on top and beginning to bubble, about 50 minutes. Remove and set aside to cool. If you must serve it the day you’ve made it, set it aside to rest for 15 minutes before slicing. For best results, allow the lasagna to cool completely and refrigerate overnight. The next day, reheat at 350 degrees, covered loosely with foil, until bubbling. Remove from heat and rest 20 minutes before serving.