Add the olive oil to a large skillet and heat it over medium-high heat.
Add the beef to the hot oil and cook, breaking it up as it cooked, until well-browned, about 6-10 minutes.
Add the onion, stirring and cooking until soft and translucent, about 3-4 minutes.
Stir in the garlic and cook until fragrant, about 1 minute.
Deglaze the pan with the red wine.
Add the bay leaves, the oregano, the cloves, the cinnamon, the allspice, the salt, the pepper, the tomato paste, the crushed tomatoes, the water, and the sugar, stirring everything together.
Bring the mixture to a boil, then reduce the heat to low.
Simmer the mixture, stirring occasionally, until the mixture has thickened, about 45 minutes.
Remove and discard the bay leaves and the cinnamon stick.
Cook the pasta according to the package directions until al dente.
Drain the pasta and allow it to cool.
In a large saucepan, melt the butter over medium heat.
Add the flour and whisk, continuing to whisk until golden brown, about 2-3 minutes.
Add the milk, the nutmeg, the salt, and the pepper, whisking everything together.
Cook, whisking as necessary, until fully combined and slightly thickened, about 5 minutes.
Transfer the saucepan from the heat and stir in 3/4 cup of the kefalotyri cheese.
In a medium-sized bowl, whisk the 2 eggs and the 2 egg yolks.
Temper the eggs by slowly ladling in some of the hot béchamel sauce, a little at time, and constantly whisking. The goal is to slowly incorporate the sauce into the egg mixture and bring the eggs slowly up to a higher temperature so they do not curdle. Do not stop whisking during this step.
Once the eggs have been tempered, gradually add the egg mixture to the béchamel sauce, stirring brisky.
Preheat the oven to 350 degrees F.
Grease a 9x13x3-inch pan with olive oil.
Toss the cooked pasta with 1/2 cup of the meat sauce.
Whisk the 2 egg whites and 1/2 cup of the kefalotyri cheese into the pasta mixture.
Transfer the pasta mixture to the prepared pan, arranging the pasta so it all goes in the same direction if desired.
Top the pasta with the remaining meat sauce, smoothing out the surface.
Pour the béchamel sauce over the meat sauce layer, smoothing it out.
Top the dish with the remaining cheese.
Bake until the top is golden-brown and set, about 45-50 minutes.
Let the pastitsio rest for at least 30 minutes.
Slice and serve.