To make the tomato sauce: Put a large skillet over medium heat and add 2 tablespoons of the olive oil. Add the onion, peppers, celery, and carrot, and cook, stirring with a wooden spoon, until the vegetables are soft, about 4 minutes. Add the garlic and continue cooking for 1 minute more. Add the water, cover, and simmer for 5 minutes. Remove the cover and cook for another 5 minutes, or until most of the water has evaporated. This will bring out the natural sugars of the vegetables.
Spray a large Dutch oven with cooking spray and put it over medium-low heat. Spoon the vegetables into the Dutch oven. To the skillet, add the remaining tablespoon olive oil. Add the ground beef and sausage, breaking it up with the wooden spoon, and cook until no pink remains. Do not let the meat mixture crisp or burn. Empty it into the Dutch oven and pour over the crushed tomatoes, quartered tomatoes, tomato paste, salt, black pepper, cayenne pepper, brown sugar, bay leaves, parsley, thyme, oregano, and vinegar. Cover and bring the sauce to a boil. Reduce the heat to low and simmer for 1 1/2 hours, stirring occasionally with the wooden spoon so that the sauce does not catch. Remove the cover and simmer for an additional 1/2 hour or until the sauce is thickened. The sauce will continue to thicken when taken off the heat. Remove the bay leaves and carefully transfer the sauce into containers. Allow the sauce to cool and refrigerate or freeze until needed. This recipe yields about 10 cups sauce.
For the ricotta-pesto layer: In the bowl of a food processor fitted with the metal blade, add the Parmesan, ricotta, pesto, black pepper, and nutmeg. Process until combined. Add the eggs and pulse until combined. Remove blade from the processor and fold in the pine nuts. Place the cheese mixture into a container and refrigerate until ready to use. Let it come to room temperature before assembling the Lasagna. This recipe yields about 5 1/2 cups cheese filling.
For the meatballs: In a large bowl, add the ground beef, sausage, 1/2 cup plus 2 tablespoons bread crumbs, cheese, onion, salt, pepper, parsley, egg, and chicken broth. Wearing disposable gloves, mix everything together well and form into tiny meatballs about 1-inch in diameter. Do not overwork or the meatballs will be tough. Roll the meatballs in the remaining 1/2 cup bread crumbs and set aside.
In a large skillet over medium heat, add the olive oil. Do not overcrowd the meatballs in the pan. When the oil is hot, fry the meatballs in batches, until they are browned and a bit crunchy on the outside, about 4 to 6 minutes. The meatballs do not need to be completely cooked through, they will finish cooking in the lasagna. Drain the meatballs on a paper towel-lined plate. Let them cool and place them in containers, and refrigerate until you are ready to assemble the lasagna. This recipe yields about 35 to 38 meatballs.
Spray the dull side of a piece of aluminum foil large enough to cover the lasagna with cooking spray so the cheese won't stick to the foil. Cover the lasagna, sprayed-side down, and put the lasagna on the prepared baking sheet. Put the baking dish in the oven and bake for 30 minutes. Uncover and bake the lasagna for another 30 minutes, or until the cheese is melted and the lasagna starts to bubble. Replace the foil if the cheese starts to brown too quickly. A thermometer placed in the center of the lasagna should read 165 to 170 degrees F when done. Remove the lasagna from the oven and place on rack to cool. Let rest for 20 minutes, if serving immediately.
If serving another day, when the lasagna is completely cool, place a sheet of paper towel on the surface, cover with plastic wrap, and refrigerate. To serve, remove the plastic and paper towel and cover with foil. Heat the lasagna in a preheated 350 degree F oven, until warmed through.
Serve the lasagna with any leftover tomato sauce and meatballs.
To assemble the lasagna: If using the fresh pasta sheets, fill a baking pan large enough to accommodate the pasta sheets with boiling water. Fill a second large pan with cold water. Immerse one pasta sheet at a time into the boiling water for 45 seconds. Remove the pasta sheet with the handle of a wooden spoon and immerse the pasta sheet immediately in cold water for 45 seconds. Remove the pasta from the cold water, shake off the excess water, and stack between layers of plastic wrap. If you are using the dry noodles, prepare according to package directions, drain, and rinse in cold water. Stack the pasta between layers of plastic wrap.
Preheat the oven to 375 degrees F. Coat a 9-by-13-inch baking dish with cooking spray.
Cover a baking sheet with aluminum foil, shiny-side up, and coat with cooking spray.
Spread 3 cups of the prepared tomato sauce over the bottom of the prepared baking dish. Place a layer of the cooked pasta on top. Spread 2 1/2-cups of the ricotta-pesto mixture over the pasta and top with 6 slices provolone cheese, overlapping them.
Place another layer of pasta on top of the provolone. Spread 2 1/2-cups of the tomato sauce on top of pasta. Set the meatballs in one layer into the sauce. Sprinkle with 1/2-cup Parmesan cheese.
Place another layer of pasta on top of the meatball layer. Spread 3-cups of the ricotta-pesto mixture on top of pasta. Sprinkle 1/2-cup Parmesan on top.
Place the last layer of pasta over the ricotta-pesto layer. Spread 3-cups tomato sauce over the top and sprinkle with remaining 1/2-cup Parmesan. Place the remaining 6 slices provolone cheese over the Parmesan, overlapping the slices.