In a 10 to 12-inch braising pan, or a Dutch oven, heat 4 tablespoons olive oil over a high flame until hot but not smoking. In a large mixing bowl, combine the ground meat, egg, parsley, bread crumbs, salt and pepper, to taste, and 2 tablespoons of grated Parmigian-Reggiano. Using a wooden spoon, mix until well combined. Using your hands, mould the meat mixture into small meatballs, each a bit larger than a golf ball. Continue making meatballs until the mixture has all been used. Add the meatballs to the pan and cook until the meatballs are browned on all sides, about 7 minutes. Add the tomatoes sauce and season, to taste, with salt and pepper. Bring the mixture to a boil then reduce to a low simmer. Cover and let simmer until the meat is tender, about 45 minutes. During this time, the pan will begin to dry out. Add the beef stock bit by bit when necessary to keep the pan moist.
Bring 6 quarts water to a rolling boil and add 2 tablespoons salt.
Roll the pasta out to the thinnest setting. Cut the resulting sheet into 4 equally-sized squares, each about 4-inches square. Carefully drop the squares into the boiling water, and cook until tender but al dente, about 5 minutes. Drain and lay gently aside.
Remove the saucepan or Dutch oven from the heat. Remove the cooked meat and set aside until cool enough to handle.
Preheat the oven to 350 degrees F.
Using the remaining teaspoon of olive oil, lightly grease an 8-inch lasagne pan. Cover the bottom of the pan with 2 sheets of cooked lasagne noodles. Cover with half of the chopped mozzarella, then half the prosciutto, then half the meatballs with some sauce. Sprinkle with half the remaining Parmigiano. Repeat this layering process using the remaining ingredients. You will end up with Parmigiano on to. Add more if desired.
Place in the oven and cook until the lasagne is heated through and the Parmigiano is brown and bubbly, about 15 minutes.
In a 3-quart saucepan, heat the olive oil over medium heat. Add the onion and garlic and cook until soft and light golden brown, about 8 to 10 minutes. Add the thyme and carrot and cook 5 minutes more, until the carrot is quite soft. Add the tomatoes and juice and bring to a boil, stirring often. Lower the heat and simmer for 30 minutes until as thick as hot cereal. Season with salt and serve. This sauce holds 1 week in the refrigerator or up to 6 months in the freezer.