Start by transferring the olive oil and ground beef to a deep pot over high heat. Avoid stirring initially to brown the meat nicely. Once browned, break apart any lumps with a wooden spoon. Cook until all juices evaporate before adding grated onion and carrot. Stir for about five minutes to soften the vegetables. Season with bay leaves, cinnamon stick, cloves, salt, black pepper, apple vinegar, pureed tomatoes, and tomato paste. Bring to a boil before reducing heat to simmer until most liquids evaporate.

While the meat sauce simmers, bring a large pot filled with salted water to boil. Add in the pastitsio pasta and cook until al dente—about three minutes less than suggested on the package. After draining, briefly rinse under cold water to stop further cooking.

In a greased casserole dish, transfer about two-thirds of the cooked pasta. Sprinkle half of your grated cheese over this layer. Spoon the prepared meat sauce over the pasta, then add the remaining pasta on top.

For the béchamel topping, heat olive oil and flour together in another pot over medium/high heat while stirring constantly until you smell nuttiness. Gradually pour in whole milk while whisking continuously to avoid lumps. Return to medium heat until thickened, then remove from heat after bubbling begins. Let cool slightly before mixing in nutmeg, salt, grated cheese, and each egg one at a time, ensuring thorough mixing after each addition.

Pour the béchamel over the layered pasta mixture, smoothing out with a spatula. Sprinkle any leftover grated cheese on top. Preheat your oven to 355°F (180°C) and bake for about fifty to sixty minutes until golden brown with dark spots. Allow resting for thirty minutes before slicing.