Prep the veggies: Roughly chop the carrot, celery, onion, garlic, and mushrooms. Pulse in a food processor until they’re about the size of the ground meat.
Brown the meat: In a large skillet over medium-high heat, cook the beef and sausage with the vegetables until browned and cooked through.
Build the sauce: Stir in the canned tomatoes, tomato paste, and Italian seasoning. Let the mixture simmer for 5–10 minutes, breaking up the tomatoes as it cooks.
Cook the pasta: Fill the empty tomato can with water and add it to the skillet along with the pasta. Simmer until the pasta is just al dente, adding more water if needed.
Mix the cheese: In a small bowl, combine the ricotta or cottage cheese with the remaining Italian seasoning.
Finish: Sprinkle the mozzarella over the lasagna, add dollops of the cheese mixture, and finish with Parmesan. Stir gently until the cheese is melted and everything is creamy.