Heat oven to 350°F. Grease a 9×13-inch baking dish.
Warm olive oil in a large pot over medium heat. Soften onion, then stir in garlic.
Add ground beef and lamb. Cook, breaking up, until browned with little pink remaining.
Stir in tomato paste and cook one minute. Splash in wine and reduce briefly.
Add crushed tomatoes, cinnamon, allspice, oregano, sugar, salt, and pepper. Simmer until thick, 20–25 minutes.
Boil pasta in salted water until just shy of al dente. Drain and toss with melted butter.
Make béchamel: Melt butter in a saucepan. Whisk in flour and cook two minutes.
Slowly whisk in warm milk. Cook, stirring, until smooth and thick enough to coat a spoon.
Season with nutmeg, a pinch of salt, and pepper. Cool one minute off heat.
Temper eggs with a little hot sauce, then whisk into béchamel. Stir in grated cheese.
Assemble: Spread half the pasta in the dish. Spoon over all meat sauce. Add remaining pasta.
Pour béchamel evenly on top. Sprinkle with the reserved cheese.
Bake until bubbling and deeply golden, 45–50 minutes. Rest 20 minutes before slicing.