Soak the Pasta: Set out a 9X13 inch (3 quart) baking dish. (Use a deep-dish pan if you have one.) Place the bucatini in the dish and cover with very hot tap water. Add a teaspoon of salt and mix the pasta around so it doesn’t stick together. Then let it soak for 30-40 minutes while the sauces are cooking.
Preheat the oven to 350 degrees F. Chop all produce.
Make the Meat Sauce: Set a large 6 quart pot over medium heat. Add the butter, onions and garlic. Sauté for 3-5 minutes, then add in the ground beef, cinnamon, nutmeg, and 1 teaspoon salt. Break the beef into small pieces with a wooden spoon as you let it brown. Once the ground beef is cooked through, add in the tomato sauce. Simmer for 15 minutes, stirring regularly. Taste, then add salt and pepper as needed.
Make the Bechamel: Set a separate pot over medium heat. Add the butter and flour. Whisk well as the butter melts to remove all clumps. Cook the roux for 1-2 minutes, do not let it brown. Once it thickens, whisk in the milk, salt, and nutmeg. Stir and let the bechamel thicken into a gravy-like consistency. Then turn off the heat. Whisking vigorously, add in the egg yolks. Whisk well so they fully incorporate. Finally stir in 1 cup of shredded cheese blend. Stir until the cheese has melted and the bechamel is smooth.
Drain and Prep the Pasta: Carefully drain the salt water off the pasta. It’s ok if a little water is left in the dish. Then toss the soaked pasta with the egg whites and feta cheese. Spread them pasta out in an even layer. Then sprinkle the top of the pasta with 2 cups of shredded cheese.
Layer All: Next spread the meat sauce over the cheese layer in the baking dish. Then gently spread the bechamel over the meat sauce. Finally sprinkle the last 1 cup of shredded cheese over the bechamel.
Bake: If you are using a standard 9X13 inch dish, set it on a rimmed baking sheet. (No need for the baking sheet if you have a deep-dish pan.) Bake uncovered, on the low rack, for 30-40 minutes. The top should be golden brown. If it starts to turn dark too quickly, cover loosely with a piece of foil.
Rest: Let the Pastitsio rest for at least 20 minutes before cutting and serving.
This lasagna will keep well in the fridge for 3-4 days. Allow leftovers to cool completely before transferring to an airtight container and refrigerating.You can easily freeze Greek lasagna before or after baking! To freeze before baking I recommend preparing the dish in a freezer-safe pan that you can then heat in the oven when ready to serve. If freezing leftovers, allow them to cool and transfer to a freezer-safe container that seals. Freeze Pastitsio for up to 2 months. Defrost in the fridge overnight before heating per the recipe instructions.