For the eggplant and potatoes

• 3 large eggplants, sliced into ¼-inch rounds
• 2 medium russet potatoes, peeled and sliced into ⅛-inch rounds
• ¼ cup olive oil, divided
• 1½ teaspoons kosher salt, plus more for salting eggplant
• ½ teaspoon black pepper

For the meat sauce

• 2 tablespoons olive oil
• 1½ pounds ground beef, ground lamb, or a combination
• 1 large yellow onion, finely diced
• 4 cloves garlic, minced
• 3 tablespoons tomato paste
• ½ cup dry red wine or beef broth
• 1 can (28 ounces) crushed tomatoes
• 1 tablespoon dried oregano
• 1 tablespoon dried mint
• ¾ teaspoon kosher salt
• ½ teaspoon black pepper
• ½ tablespoon ground cinnamon
• ¼ teaspoon ground allspice
• 1 bay leaf
• ¼ cup chopped fresh parsley

For the béchamel

• 6 tablespoons butter
• ½ cup all-purpose flour
• 4 cups whole milk, warmed
• 2 egg yolks
• ¾ cup finely grated Parmesan cheese
• ¼ teaspoon ground nutmeg
• ¾ teaspoon kosher salt
• ¼ teaspoon black pepper