Heat the onions, celery and carrots in olive oil in a large pan over a medium-low flame. Season with salt and cook until soft and tender, about 15 minutes. Add the garlic and cook until fragrant, about 1 minute. Add the fresh basil and cook until you start to smell its essence. Add the ground beef and pork. Turn up the flame to medium and cook the meat just until it is kissed by the heat and begins to brown, about 5 minutes.
Once the beef and pork begins to release its juices, add the wine and simmer until the alcohol aroma dissipates and the liquid is reduced by about ⅓, about a minute. Remove and discard the garlic. Add the tomatoes, pressing them into the meat with the back of a wooden spoon or with a potato masher until they are incorporated with the meat sauce. Turn the flame down to low and simmer, covered, for 1 ½ hours.
Make The Sfoglia all’Uovo
Pour the flour onto your work surface and use your fingers or a spoon to make a well in the center. Crack the eggs into the well and gently beat them together with a fork. Little by little and working from the inside of the well to the edges, bring some flour into the egg mixture, stirring with a fork until each portion of flour is incorporated. When the egg mixture is no longer runny, begin kneading the dough with your hands, feeling how it changes beneath your fingertips as you work.
Use your whole body to rock back and forth over the dough, pressing it against your work surface until the eggs and flour have become one. Continue kneading to strengthen the dough, working it over like a lover, until it is strong and glossy and it springs back when you press a finger into it. Wrap tightly with plastic wrap and set aside for 30 minutes.
Unwrap the pasta dough, cut it into 4 pieces and flatten each to a uniform thickness. Run each dough piece through a pasta machine, starting on the widest setting and repeating on successively thinner settings until it is almost transparent. Pasta machines typically make long sheets 6 inches wide. Cut into sheets to a length that fits your lasagna pan well.
Make The Bechamel
In a small saucepan, heat the milk over medium-low heat.
In a separate small saucepan, melt the butter over medium heat until frothy. Whisk in the flour with the passion of an Italian until the mixture is lightly browned and smells nutty, about 3 minutes. Add the warmed milk and continue whisking until the mixture thickens enough to coat the back of a spoon, about 3 minutes longer. Season with the nutmeg and salt to taste.
Assemble the Lasagna
Preheat the oven to 375ºF. Grease a 9- x 13-inch baking dish with butter.
Spoon a ½ cup layer of ragù on the bottom of the baking dish. Place a pasta sheet on top. There’s no need to cook the pasta in advance. It will cook in the juices of the ragù. Spoon another ½ cup of ragù over the pasta. Distribute ½ cup of bechamel, and ½ cup of Parmigiano-Reggiano evenly over the tray. Repeat the pasta, ragù, béchamel, and Parmigiano-Reggiano layering until all of the pasta sheets have been used. The top layer should be Parmigiano-Reggiano .
Cover the baking dish and bake for 40 minutes. Allow the lasagna to rest for about 30 minutes before serving in slices.