Adjust the oven rack to the middle position and heat to 425°F. Toss the eggplant with oil, salt, and pepper on two rimmed baking sheets. Roast until tender and lightly browned, about 30 minutes, switching and rotating the sheets halfway through.

While the eggplant roasts, heat a splash of oil in a Dutch oven over medium heat. Sauté the onion until softened and translucent, about 5 minutes. Stir in the garlic, oregano, and cinnamon; cook until the spices are fragrant and toasted.

Pour in the wine and simmer, scraping up any browned bits from the bottom of the pot, until the liquid has reduced by half.

Stir in the potatoes, broth, and tomatoes. Bring the mixture to a simmer and cook until the potatoes are tender when pierced with a fork, about 15 minutes.

Stir in the bulgur, remove the pot from the heat, and cover. Let the mixture sit until the bulgur has absorbed the excess liquid and softened, about 15 minutes.

For the béchamel, melt the butter in a medium saucepan over medium heat. Whisk in the flour and cook for 1 minute, then slowly whisk in the milk in a steady stream. Simmer, whisking constantly, until the sauce thickens enough to coat the back of a spoon; stir in the Parmesan and nutmeg.

Spread the potato-bulgur mixture into a 13 by 9-inch baking dish and smooth into an even layer. Arrange the roasted eggplant over the top, then pour the béchamel over the eggplant layer.

Bake until the sauce is golden brown and bubbling, about 25 minutes. Let the dish rest for at least 15 minutes before serving — this allows the layers to firm up for clean slicing.

Scatter the fresh basil over the top and serve warm.