In a large skillet over medium heat, heat oil. Add onion, garlic, and ½ teaspoon salt and cook, stirring occasionally, until softened, 4 to 5 minutes. Add beef and cook, stirring to break up large clumps, until browned and just cooked through, 4 to 5 minutes. Drain any excess fat.

Stir in tomato paste and cook, stirring occasionally, until paste is incorporated and beginning to stick to pan, 1 to 2 minutes. Add wine and cook, scraping up any browned bits, until pot is almost dry, 1 to 2 minutes. Stir in tomatoes, stock, bay leaf, cinnamon, cloves, and remaining 1 ½ teaspoons salt. Bring to a boil over medium-high heat, then reduce heat to medium-low and simmer, stirring occasionally, until reduced to a thick sauce, about 45 minutes.

Remove from heat. Discard bay leaf and let sauce cool to room temperature.

Make Ahead: Sauce can be made 3 days ahead. Let cool completely, then store in an airtight container and refrigerate.

Preheat oven to 350°. Grease a 13"-by-9" casserole dish with cooking spray.

Bring a large pot of water to a boil over high heat. Add bucatini and 2 tablespoons salt and cook, stirring occasionally, until al dente, 7 to 8 minutes (pasta should still be slightly undercooked in the center).

Drain pasta and transfer to a large bowl. Add feta and 2 egg whites and toss to combine. Arrange pasta mixture in bottom of prepared dish, encouraging strands to lay as straight as possible, and top with meat sauce; set aside.

In a medium pot over medium heat, melt butter. Stir in flour and cook, stirring constantly, until mixture foams, 2 to 3 minutes. Whisk in milk. Increase heat to medium-high and cook, stirring and constantly scraping bottom of pot, until thick and velvety, 2 to 3 minutes more. Remove from heat and add nutmeg, 6 ounces Kefalotyri cheese, and remaining 1 teaspoon salt. Stir until cheese is melted, then whisk in 2 egg yolks.

Spoon béchamel sauce over top of meat sauce; sprinkle with remaining 2 ounces Kefalotyri cheese.

Bake until bubbling around the edges and browned on top, 45 to 55 minutes. Let cool 15 minutes before serving.