In a deep pot, heat olive oil over medium heat and cook ground beef until browned, about 5-7 minutes. Add chopped onion and grated carrot, cooking until softened, around 3-4 minutes. Mix in bay leaves, cinnamon stick, ground cloves, salt, and pepper. Pour in dry wine to deglaze and simmer until reduced, about 2 minutes. Mix in pureed tomatoes and tomato paste; simmer for about 20 minutes until thick.

While the meat sauce simmers, bring salted water to a boil in a large pot. Cook Pastitsio pasta until al dente, about 3 minutes shy of package instructions. Drain and rinse under cold water. Mix pasta with grated cheese in a greased casserole dish.

In a separate saucepan, heat olive oil and whisk in flour for about 1-2 minutes. Gradually whisk in milk, stirring until the mixture thickens, around 5-7 minutes. Stir in nutmeg, salt, grated cheese, and incorporate eggs one at a time.

Preheat oven to 375°F (190°C). In the casserole dish, layer half of the pasta as the base, spoon over meat sauce, add remaining pasta, and pour béchamel sauce on top, spreading evenly. Sprinkle additional cheese.

Bake for 50-60 minutes until bubbly and golden brown. Let rest for 15 minutes before slicing.