Set out a 9x13-inch (3-quart) baking dish (use deep-dish if available). Put the bucatini or penne in the dish and cover with very hot tap water; add 1 teaspoon salt and stir so the pasta doesn't clump. Let soak 30–40 minutes while you make the sauces.
Preheat the oven to 350°F (175°C). Chop the onion and mince the garlic.
Make the meat sauce: In a large pot over medium heat, melt 2 tablespoons butter. Add the chopped onion and minced garlic and sauté 3–5 minutes until softened. Add the ground beef, 1/4 teaspoon cinnamon, 1/8 teaspoon nutmeg, and 1 teaspoon salt; break the beef into small pieces and cook until no longer pink.
Stir the 30 ounces tomato sauce into the browned beef. Reduce heat and simmer about 15 minutes, stirring occasionally. Taste and adjust salt and pepper as needed.
Make the béchamel: In a separate saucepan over medium heat, melt 1/2 cup butter. Whisk in 1/2 cup flour and cook 1–2 minutes without browning to make a roux. Gradually whisk in the 4 cups milk and continue stirring until the sauce thickens to gravy consistency.
Remove the béchamel from the heat. Vigorously whisk in the 2 reserved egg yolks until smooth, then stir in 1 cup of the shredded Italian blend cheese until melted. Season with 1 teaspoon salt and 1/8 teaspoon nutmeg.
Drain most of the soaking water from the pasta (it's okay if a little remains). Toss the pasta with the 2 large egg whites and the 6 ounces crumbled feta until evenly coated.
Spread the pasta in an even layer in the prepared baking dish. Sprinkle 2 cups of the shredded Italian blend cheese over the pasta.
Spoon the meat sauce over the cheese layer and spread gently to an even layer. Pour or spread the béchamel over the meat sauce, smoothing the top. Sprinkle the remaining 1 cup shredded cheese over the béchamel.
If using a standard 9x13-inch dish, place it on a rimmed baking sheet; otherwise skip the sheet for a deep-dish pan. Bake uncovered on the low oven rack 30–40 minutes until the top is golden. If it browns too quickly, cover loosely with foil.
Let the pastitsio rest at least 20 minutes before cutting and serving.