Place a rack in middle of oven; preheat to 350°.

Heat 1 Tbsp. extra-virgin olive oil in a large Dutch oven or other heavy pot over medium-high.

Cook 1 lb. ground beef (20% fat) and 8 oz. hot Italian sausage, casings removed, breaking up into small pieces with a wooden spoon, until well browned and cooked through, 10–12 minutes.

Using a slotted spoon, transfer meat to a medium bowl, leaving fat behind.

Add 2 large onions, finely chopped, 6 garlic cloves, finely chopped, and 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt to same pot; if pot looks dry, add up to 1 Tbsp. more oil.

Cook, stirring often, until onions are translucent and golden around the edges, 8–10 minutes.

Add 2 Tbsp. double-concentrated tomato paste and 1 Tbsp. dried oregano; cook, stirring constantly, until paste is brick red and beginning to stick to bottom of pot, 5–8 minutes.

Return meat to pot.

Add one 28-oz. can plus one 14.5-oz. can crushed tomatoes, ¼ cup Worcestershire sauce, 1 tsp. sugar, and 1 tsp. freshly ground pepper to pot.

Bring to a simmer, stirring occasionally and scraping up any browned bits stuck to bottom of pot.

Remove from heat and stir in 2 oz. Parmesan, finely grated (about 1 cup).

Taste tomato sauce and season with more salt if needed.

Toss 12 oz. Fontina cheese, grated (about 3 cups), and remaining 4 oz. Parmesan, finely grated (about 2 cups), in a large bowl to combine; measure out 1 cup cheese and set aside for topping.

Add 2 large eggs, 24 oz. whole-milk cottage cheese, and remaining 2 garlic cloves, finely chopped, 2 tsp. Diamond Crystal or 1¼ tsp. Morton kosher salt, and 1 tsp. freshly ground pepper to remaining cheese in bowl.

Mix vigorously with a wooden spoon until combined.

Arrange 3 regular dried lasagna noodles in a single layer in a 13x9" baking dish.

Spread 1½ cups tomato sauce over noodles.

Scatter one third of Fontina mixture evenly over sauce.

Starting with another layer of noodles, repeat process, creating 4 layers of noodles in total and ending with a layer of tomato sauce.

Top lasagna with reserved cheese.

Place baking dish on a large rimmed baking sheet (to catch drips) and bake lasagna until sauce is bubbling around the edges, top is golden brown, and noodles are al dente, 75–85 minutes.

Let sit at least 15 minutes and up to 45 minutes before serving.

Do Ahead: Lasagna can be baked 2 days ahead. Let cool completely; cover loosely with foil and chill. Reheat, covered, in a 325° oven until warmed through, 30–45 minutes.