For the sauce, heat a large, heavy-bottomed pot over medium-high heat. Add the oil and heat until it is shimmering, almost smoking. Add the beef, pork, veal and sausage; and cook until nicely browned, 10-15 minutes, using a wooden spoon or fork to break up and blend the meats. Stir in the garlic and oregano, and cook, stirring, until the garlic is fragrant, 1-2 minutes. Stir in the tomato paste and cook, stirring, until the meat is coated with the paste, about 2 minutes. Pour in the tomatoes with their juices and the tomato purée, season with salt and pepper, and bring to a boil. Lower the heat and simmer over medium heat, stirring occasionally, until the sauce is thickened, about 1 1/2 hours. Remove from the heat and let cool. The sauce can be refrigerated in an airtight container for up to 2 days or frozen for up to 1 month. Let it come to room temperature before making the lasagna.
Put the ricotta, parsley, basil, 1 cup of the Parmigiano-Reggiano, 1 pound of the shredded mozzarella and the eggs in a large bowl. Season with salt and pepper, and blend well.
Cook the lasagna noodles, if necessary. (Fresh pasta and no-cook lasagna do not require any precooking. Dry, curly edged lasagna noodles need only be par-cooked for 3 minutes in boiling, salted water. Drain, rinse in cold water and pat dry with paper towels.
Position a rack a rack in the center of the oven. Preheat the oven to 375° F.
Spread 1 cup of the sauce in the bottom of a 9x13x2” glass or ceramic baking dish. Line the bottom of the dish with overlapping noodles. Spread 1/3 of the cheese mixture over the noodles, sprinkle with a little Parmigiano-Reggiano and shredded mozzarella, then top with 1 1/2 cups of the sauce and another layer of noodles. Repeat this layering sequence twice, finishing with a layer of pasta and 1 1/2 cups of sauce. (There may be some sauce left over, save it for another use.)
Put the remaining 1/2 cup of mozzarella and 1/2 cup of Parmigiano-Reggiano in a small bowl, stir together, and scatter evenly over the lasagna.
Cover the lasagna with aluminum foil and bake for 40 minutes. (Put the a rimmed cookie or baking sheet into the oven under the lasagna pan to catch any spills and boil-over.) Remove the foil and continue bake until the top is golden and crisp around the edges, and the filling is bubbling, about another 20 minutes. Remove the lasagna from the oven and let rest for 20 minutes before slicing and serving.