2 cups plus 3 tbsp. extra-virgin olive oil
6 bay leaves
2 sticks cinnamon
1 large yellow onion, finely chopped
2 lb. ground beef
1⁄4 cup tomato paste
1 1⁄2 tsp. ground cinnamon, plus more for dusting
1⁄2 tsp. freshly grated nutmeg
1⁄4 tsp. ground cloves
1 tbsp. red wine vinegar
1 tsp. sugar
1 1⁄2 tsp. kosher salt, plus more to taste
1 (28-oz.) can whole peeled tomatoes in juice, crushed
8 tbsp. unsalted butter
1 cup flour
4 cups milk
4 eggs, beaten
3 large eggplant, cut into 1/4″-thick slices
5 medium russet potatoes, peeled, thinly sliced crosswise
1 cup coarsely grated graviera or Gruyère
Freshly ground black pepper, to taste
Heat 3 tbsp. oil in a 6-qt. saucepan over medium-high heat. Add bay leaves, cinnamon sticks, and onion; cook until soft, about 5 minutes. Add beef; cook until all liquid evaporates and meat is browned, about 30 minutes. Add tomato paste, 1 1⁄4 tsp. ground cinnamon, 1⁄4 tsp. nutmeg, and cloves; cook until lightly caramelized, about 2 minutes. Add vinegar, sugar, tomatoes, and 2 cups water; boil. Reduce heat to medium-low; cook, covered partially, until almost all liquid is evaporated, about 1 1⁄2 hours. Remove from heat; discard cinnamon and bay leaves. Season with salt and pepper; set meat sauce aside. Heat butter in a 2-qt. saucepan over medium-high heat. Add flour; cook until smooth, about 2 minutes. Add 1 1⁄2 tsp. salt, remaining cinnamon and nutmeg, and milk; cook until thickened, about 10 minutes. Remove from heat; pour into a blender with eggs. Blend until smooth; set béchamel aside.
Heat oven to 350°. Heat remaining oil in a 12″ skillet over medium-high heat. Dust eggplant with cinnamon; working in batches, fry in oil until golden brown, about 10 minutes. Transfer to paper towels to drain; set aside. Bring a large pot of salted water to a boil. Add potatoes; cook until barely tender, about 10 minutes. Drain; transfer to a bowl of ice water to stop cooking. Drain; set aside.
Spread 1 cup béchamel on bottom of a 10″ x 14″ baking dish; sprinkle with 1⁄3 cup graviera. Spread potatoes over cheese; top with eggplant. Pour meat sauce over eggplant; spread remaining béchamel over meat sauce. Sprinkle with remaining graviera; bake until golden brown, about 1 hour.