Put the eggplant in a colander set over a bowl, sprinkle generously with kosher salt, and set aside.
In a 12-inch skillet, heat 2 Tbs. of the olive oil over medium-low heat. Add the onion and cook, stirring occasionally, until softened, about 10 minutes.
Raise the heat to medium high and add the lamb, stirring until it browns lightly and begins to stick to the pan, about 6 minutes. Lower the heat and add the garlic, tomato paste, cinnamon, and 1 tsp. sea salt. Pour in the sauce and stir to deglaze the pan. Cook, stirring frequently, until the liquid is slightly reduced but the lamb is still moist, 1 to 2 minutes. Stir in the parsley and season with pepper. Transfer to a large bowl and set aside.
Wash and dry the skillet. Pat the eggplant dry with paper towels. Heat the remaining 2 Tbs. olive oil in the pan over medium-high heat. When hot, add the eggplant and cook, stirring occasionally, until lightly browned in spots, about 2 minutes. Reduce the heat to medium low, cover, and cook until soft, about 10 minutes.
Position a rack in the center of the oven and heat the oven to 400°F.
Uncover and stir the eggplant with a wooden spoon, scraping the browned bits from the bottom of the pan. Remove from the heat and mash the eggplant with the back of the spoon. Stir the eggplant into the lamb mixture and season to taste with salt and pepper. Spoon the mixture into six 8-oz. ramekins. (French onion soup bowls would work, too.)
Put the milk in a 2-quart saucepan with the bay leaf and mace. Bring to a boil over medium heat. Remove from the heat, cover, and let sit for 10 minutes to infuse the milk. Strain into a liquid measuring cup and set aside.
In a 4-quart saucepan, melt the butter over medium heat. Add the flour, stirring constantly, and cook until the mixture is lightly colored, about 2 minutes. Whisk in the reserved milk and cook, whisking constantly, until thickened and shiny, 3 to 4 minutes. Put the egg yolk in a small bowl and whisk with about 1/4 cup of the sauce. Add the yolk and sauce back to the saucepan and whisk to combine. Whisk in the cheese. Remove from the heat and let cool slightly.
In a small bowl, whisk the egg white until stiff peaks form and then fold the white into the cooled sauce. Spoon the sauce over the top of the eggplant and lamb mixture, dividing evenly.
Put the ramekins in a 4-quart, 15x10-inch glass baking dish, add 1 cup of water to the dish, cover with foil, and bake for 15 minutes. Uncover and continue baking until bubbling and browned, about 35 minutes more. Let cool for 10 minutes and serve.