In a large skillet, heat olive oil over medium heat and sauté chopped onions, diced carrots, and celery for about 5 minutes until softened. Stir in the ground beef and Italian sausage, cooking until browned (about 7-10 minutes). Add minced garlic, dried oregano, and red pepper flakes for 1 minute. Mix in tomato paste and puréed tomatoes, and simmer for 15 minutes.

In a mixing bowl, combine ricotta cheese, a splash of milk, beaten egg, grated Parmesan, shredded mozzarella, and parsley. Stir until creamy and well-integrated.

In a 9x13-inch baking dish, spread a thin layer of meat sauce on the bottom. Cover with a layer of dry lasagna noodles, spread half of the ricotta mixture, and add more meat sauce. Repeat layering until finished with sauce on top.

Sprinkle remaining mozzarella and Parmesan over the top. Cover with foil and bake for 45 minutes covered at 375°F. Remove foil and bake for an additional 20 minutes until golden brown.

Allow lasagna to rest for 10-15 minutes after baking, then slice and serve.