Mound 3 1/2 cups of the flour in the center of a large wooden cutting board. Make a well in the middle of the flour and add the eggs, spinach, and the olive oil. Using a fork, beat together the spinach, eggs, and oil, and begin to incorporate the flour, starting with the inner rim of the well.
As you expand the well, keep pushing the flour up from the base of the mound to retain the well shape. The dough will come together when half of the flour is incorporated.
Start kneading the dough with both hands, using the palms of your hands. Once you have a cohesive mass, remove the dough from the board and scrape up and discard any leftover bits. Lightly reflour the board and continue kneading for 6 more minutes. The dough should be elastic and a little sticky. Wrap the dough in plastic and allow to rest for 30 minutes at room temperature.
Divide the dough into 3 equal portions and roll each out to the thinnest setting on a pasta rolling machine.
Bring about 6 quarts of water to a boil and add 2 tablespoons salt. Set up an ice bath next to the stove top. Using a sharp knife, carefully cut the pasta into 20 (5-inch squares) and drop into the boiling water. Cook 1 minute, until tender. Drain well and refresh in the ice bath. Once cooled, drain the pasta squares on towels and set aside.
Preheat the oven to 375 degrees F.
In a brownie pan, assemble the lasagne, beginning with a layer of ragu, a sprinkling of grated Parmigiano, a layer of pasta, a layer of bechamel, and then again a layer of ragu, a sprinkling of grated Parmigiano etc. until all sauce and pasta are used up. The top layer should be pasta with bechamel over it. Top the lasagne with grated Parmigiano-Reggiano and bake in the oven until the edges are browned and the sauces are bubbling, about 45 minutes to 1 hour.
Remove and allow to cool for at least 20 minutes before serving with a dusting of Parmigiano.
In a 6- to 8-quart, heavy-bottomed saucepan, heat the olive oil and butter over medium heat. Add the onion, celery, carrot, and garlic and sweat over medium heat until vegetables are translucent. Add veal, pork, beef, and pancetta to the vegetables, brown over high heat, stirring to keep meat from sticking together for about 15 to 20 minutes. Add the milk and simmer until almost dry, about 10 minutes. Add the tomatoes and simmer 15 minutes. Add the wine and bring to a boil, lower the heat, and simmer for 2 to 2 1/2 hours, until flavors are developed. Season with salt and pepper, to taste, and remove from the heat.
In a medium saucepan, heat butter until melted. Add flour and stir until smooth. Over medium heat, cook until light golden brown, about 6 to 7 minutes.
Meanwhile, heat milk in separate pan until just about to boil. Add milk to butter mixture 1 cup at a time, whisking continuously until very smooth and bring to a boil. Cook 10 minutes and remove from heat. Season with salt and nutmeg and set aside.