Preheat oven to 375°F. Heat oil in a large saucepan over medium-high. Add beef, sausage, onion, carrots, celery, and 1 teaspoon salt. Cook, stirring occasionally and breaking up meat into bite-size pieces, until vegetables soften and meat has browned, about 10 minutes.

Add garlic, oregano, and red-pepper flakes; cook until fragrant, 1 minute.

Add tomato paste; cook 2 minutes more.

Puree tomatoes in a blender or food processor.

Add tomatoes to pan. Bring to a boil, then reduce heat to low and simmer, stirring occasionally, until reduced and thickened slightly but still very loose, about 15 minutes. Remove from heat.

In a large bowl, whisk together ricotta, milk, eggs, 1 cup Parmigiano-Reggiano, 2 ½ cups mozzarella, parsley, 1 teaspoon salt, and ¼ teaspoon pepper.

Spread 2 cups meat sauce evenly in a 9-by-13-inch baking dish, preferably ceramic or enameled cast iron.

Arrange 3 lasagna noodles over sauce (space noodles evenly; they will expand as the lasagna cooks).

Spread 2 cups cheese mixture over noodles.

Top with 3 more noodles.

Repeat layering of meat and cheese mixtures between noodles twice more.

Spread 2 cups meat sauce evenly over top layer of noodles.

Sprinkle evenly with remaining 1 ½ cups mozzarella and ¼ cup Parmigiano-Reggiano. Loosely cover with parchment part then with aluminum foil and transfer to a rimmed baking sheet.

Bake until noodles are tender but still al dente (easily pierced with the tip of a knife), 40 to 45 minutes. Uncover; continue baking until bubbly and puffed slightly along edges and golden brown in places on top, 20 to 25 minutes more.

Transfer dish to a wire rack and let cool slightly and set, about 20 minutes; cut into squares to serve.