Purée the tomatoes in a blender and set aside. Put currants into a small bowl and cover with boiling water; let soften for 30 minutes. Drain currants and set aside. Heat 1 tbsp. olive oil in a 6-qt. pot over medium-high heat. Add the lamb, cayenne, cinnamon, ginger, allspice, and salt and pepper and cook, stirring to break up the meat, until browned, about 5 minutes. Transfer lamb to a large strainer set over a bowl and drain; discard any liquid left in the pot. Return pot to the heat and add the remaining olive oil along with the garlic, onions, and bell pepper and cook, stirring occasionally, until soft, about 10 minutes. Add the wine and cook, stirring occasionally, until almost evaporated, 10-15 minutes. Add the reserved tomatoes, currants, and lamb and bring to a boil. Reduce the heat to medium-low and simmer until thickened, about 30 minutes. Remove from the heat and set meat sauce aside.
Heat the canola oil in 12" skillet over medium-high heat. Working in batches, add the eggplant slices and fry, turning occasionally, until tender, about 5 minutes. Transfer eggplant slices to paper towels. Working in batches, add the potatoes and cook until tender, about 5 minutes, and transfer to paper towels.
Make a béchamel sauce: Melt butter in a 2-qt. saucepan over medium heat. Add flour and cook, whisking constantly, until pale and smooth, 2 minutes. Whisking constantly, add the milk in a steady stream until incorporated; add the bay leaf and cook, whisking often, until reduced to 2 cups, about 15 minutes. Season with salt, pepper, and nutmeg and discard the bay leaf. Let sauce cool for 5 minutes. In a small bowl, whisk together the yogurt and egg yolks and whisk into sauce until smooth.
Heat oven to 400°. Place the reserved potato slices in the bottom of an oval 3-qt. baking dish (or two 1 1⁄2-qt. baking dishes) and season with salt and pepper. Put the eggplant slices on top, season with salt and pepper, and then cover with the meat sauce. Pour the béchamel over the top of the meat sauce and spread evenly with a rubber spatula. Sprinkle parmesan evenly over the top and bake until browned and bubbly, 45–50 minutes. Let cool for at least 20 minutes before serving.