Adjust oven racks to the upper-middle and lower-middle positions and heat the oven to 450 degrees. Line two rimmed baking sheets with foil and spray with vegetable oil spray. Toss the eggplant with oil, salt, and pepper and spread evenly over the sheets. Bake until the eggplant is light golden brown and tender, 40 to 50 minutes, switching and rotating the sheets halfway through for even cooking. Set the eggplant aside to cool and reduce the oven temperature to 400 degrees.

Heat the remaining oil in a Dutch oven over medium heat until shimmering. Add the onion and salt and cook until softened, about 5 minutes. Stir in the garlic, oregano, and cinnamon and cook until fragrant. Add the wine, scraping any browned bits from the bottom of the pot, and cook until the liquid has nearly evaporated, about 2 minutes.

Stir in the potatoes and broth and bring to a simmer. Cover the pot, reduce the heat to low, and cook until the potatoes are nearly tender, about 15 minutes. Stir in the tomatoes and their juice and cook uncovered until the flavors meld, about 5 minutes.

Remove the pot from the heat and stir in the bulgur. Let the mixture sit until the grains are tender and most of the liquid is absorbed, about 15 minutes. Transfer the mixture to a 13 by 9-inch baking dish and top evenly with the roasted eggplant to form a compact layer.

Melt the butter in the empty pot over medium heat. Stir in the flour and cook for 1 minute to remove the raw taste. Gradually whisk in the milk and bring to a simmer, whisking often, until the sauce thickens, about 5 minutes. Off the heat, whisk in the Parmesan and nutmeg, then season with salt and pepper to taste.

Pour the sauce over the eggplant and smooth it into an even layer. Cover the dish with foil and bake until the edges are bubbling, about 15 minutes. Uncover and continue to bake until the top is light golden brown, about 15 minutes more. Let the dish cool for 10 minutes to set before sprinkling with fresh basil and serving.