In a large pot, heat the olive oil over high heat. Add the minced garlic and chopped onions, cooking for 2-3 minutes until the onions soften.

Add the beef mince, breaking it up as it cooks until it turns from red to brown.

Stir in the remaining meat sauce ingredients. Bring to a simmer, then reduce the heat to medium-low. Cook for 45 minutes to 1 hour, stirring occasionally, until the mixture is thick with little liquid. Remove from heat and let cool to room temperature.

In a large saucepan, melt the butter over medium heat. Add the flour and stir for 1 minute.

Gradually pour in half of the milk while stirring to form a wet paste. Then, add the remaining milk, whisking until the mixture is lump-free.

Cook for about 5 minutes, stirring constantly, until the sauce thickens enough to coat the back of a wooden spoon. Remove from heat and stir in the nutmeg, cheese, and salt. Let cool for 5 minutes, then whisk in the egg yolks.

Cook the pasta according to package instructions, but reduce the cooking time by 1 minute. Drain and return to the pot. Let cool for 3 minutes, then stir in the egg whites and crumbled feta.

Preheat your oven to 350°F (180°C). In a baking dish (9 x 13 x 2.75 inches), layer the pasta evenly. Top with the meat sauce, smoothing it out, then pour over the Béchamel sauce. Sprinkle the grated cheese on top.

Bake for 30 minutes or until the top is golden brown. Allow to cool for at least 10 minutes before slicing to serve.