Meat Sauce In a large saucepan, heat the olive oil over medium-low heat. Add onions and cook until softened and slightly browned, about 5 minutes. Add garlic and cook 30 seconds. Add beef and increase heat to medium-high and cook, breaking up meat, until it is thoroughly browned. Stir in crushed tomatoes, tomato paste, parsley, salt, pepper, sugar, cinnamon stick, and bay leaves and bring to a boil. Reduce heat to low and simmer, uncovered, until most of the liquid has been absorbed and mixture is a thick meat reduction, about 1 hour, then remove from heat and discard the cinnamon stick and bay leaves.

Make Pasta While sauce is simmering, bring a large pot of water to a boil and cook the ziti to al dente. Drain pasta, return to the pot and immediately toss with melted butter; set aside to cool slightly.

White Sauce In a large saute pan, melt butter over medium heat. Add flour and whisk until slightly browned, about 1 minute. Gradually add milk, whisking constantly. Continue cooking and whisking until sauce is thickened, 10-15 minutes. Remove from heat and whisk in salt, pepper and nutmeg and set aside to cool slightly.

Preheat oven to 375F. Grease a 9x13 baking dish; set aside.

Assemble Pastitsio Sprinkle bottom of the greased baking dish with 2 Tbsp breadcrumbs. Stir beaten egg and cheese into pasta and spread half of pasta mixture onto bottom of baking dish. Fold egg white into meat sauce and spread entire mixture on top of the pasta. Top with remaining pasta. Whisk egg yolks and cheese into white sauce and spread on top of pasta. In a small bowl, stir together remaining ¼ cup breadcrumbs and cheese and sprinkle on top of the white sauce.

Bake until the surface of the pastitsio is browned and bubbling, 45 minutes to 1 hour. Allow to sit for 15 minutes before slicing and serving. Leftovers should be stored, covered, in the refrigerator for up to 4 days.