Position a rack in the upper third of an oven and preheat to 400°F.
In a 10-inch nonstick fry pan over medium-high heat, warm 2 tsp. of the olive oil. Add the sausage and cook, stirring and crumbling with a wooden spoon, until browned, about 5 minutes. Transfer to a paper towel-lined plate.
Discard all but 1 Tbs. of the fat in the pan. Set the pan over medium-high heat and warm 2 tsp. of the olive oil. Add the eggplant and cook, stirring occasionally, under tender and browned, about 6 minutes. Transfer to a large bowl.
Set the pan over medium heat and warm the remaining 1 Tbs. olive oil. Add the onion and cook until softened, 5 to 6 minutes. Add the garlic and cook for 1 minute. Add the wine, increase the heat to medium-high and bring to a boil, stirring to scrape up the browned bits. Cook until the liquid is almost evaporated, about 3 minutes. Add the tomatoes and their juices and simmer until thickened, about 10 minutes.
Add the sausage, pasta, basil, olives, 2 cups of the mozzarella and the tomato sauce to the bowl with the eggplant and stir to combine. Season with salt and pepper. Transfer the pasta mixture to the fry pan and sprinkle the remaining 1/2 cup mozzarella and the Parmigiano-Reggiano on top. Cover the pan, transfer to the oven and bake for 25 minutes. Uncover the pan and broil at 500°F for 5 minutes. Let stand for 10 minutes before serving