Make the sauce: Coat a large saucepot with the olive oil and add the pancetta. Put the pot over medium-high heat and cook the pancetta 4 to 5 minutes. Add the onions, season generously with salt and stir to coat with the olive oil. Cook 6 to 7 minutes, stirring; the onions should become very soft and aromatic but have no color. Add the garlic and cook for another 2 to 3 minutes, stirring frequently.
Pass the tomatoes through a food mill; be sure to scrape the pulp off the bottom of the food mill. Add the tomatoes to the pot; fill one of the empty tomato cans with water and add to the pot (2 to 3 cups). Season generously with salt and taste; tomatoes take a lot of salt. Simmer 2 to 3 hours over medium-low heat, stirring occasionally and tasting frequently. Season in baby steps, tasting every step of the way.
Make the lasagna: Bring a large pot of salted water to a boil. Working in batches, cook the lasagna noodles until pliable but not limp, 6 to 7 minutes. Remove with tongs and transfer to a bowl of ice water to cool; remove and lay flat on a sheet tray.
Coat a large saute pan with 2 tablespoons olive oil and put over medium-high heat. Add the sausage and cook, stirring, until brown and crumbly. Remove from the pan and drain on paper towels.
Ditch the fat from the pan and add 1 tablespoon olive oil along with 2 garlic cloves and a pinch of red pepper flakes. Put the pan over medium-high heat. When the garlic is golden, remove it and discard. Add the mushrooms and season with 1/2 teaspoon salt; cook until soft, 4 to 5 minutes. Remove from the pan and reserve. Repeat this process with the remaining 1 tablespoon olive oil, 2 garlic cloves and pinch of red pepper flakes, and the zucchini.
In a bowl, mix the ricotta, half of the parmigiano, the eggs, basil and 1 teaspoon salt.
Preheat the oven to 350 degrees F. In the bottom of a deep 9-by-13-inch baking dish, evenly spread a couple ladlefuls of sauce. Cover with a layer of the lasagna noodles. Spread one-third of the ricotta mixture over the pasta. Top with a layer of pasta in the opposite direction of the first layer (this will give the lasagna a little more stability). Spread a light layer of sauce on the pasta and sprinkle with one-third of the sausage; top with one-third each of the mushrooms and zucchini. Top with a layer of mozzarella and some of the remaining parmigiano. Repeat these layers until all the ingredients have been used. Be sure to finish with a layer of pasta covered with sauce, mozzarella and parmigiano. Cover with aluminum foil.
Place the lasagna on a baking sheet and bake in the oven until hot and bubbly, about 1 hour, 15 minutes, removing the foil for the last 15 minutes of cooking. Let cool 20 minutes before slicing.