1. Heat the broiler. Arrange the eggplant in a single layer on a large baking sheet. Coat both sides of the eggplant with 2 1/2 tablespoons of the oil and sprinkle with 1/2 teaspoon of the salt and 1/4 teaspoon pepper. Broil, turning once, until browned, about 5 minutes per side. Turn off the broiler and heat the oven to 425°.
2. In a large nonstick frying pan, heat 1 tablespoon of the oil over moderately high heat. Season the chicken with 1/4 teaspoon of the salt and 1/8 teaspoon pepper and add to the pan. Partially cook the chicken for 2 minutes per side and remove from the pan. When cool enough to handle, cut the chicken crosswise into 1/4-inch slices.
3. Oil an 8-inch square baking dish. Put one third of the eggplant in a single layer in the dish. Top with half of the chicken, half of the tomatoes, half of the mozzarella, one third of the Parmesan, half of the basil, and the remaining 1/4 teaspoon of salt. Repeat with another third of the eggplant, the remaining chicken, tomatoes, and mozzarella, another third of the Parmesan, and the remaining basil. Top with the remaining eggplant and sprinkle with the remaining cheese. Drizzle with the remaining 1/2 tablespoon oil. Bake for 20 minutes and let sit for 5 minutes before cutting.
Wine Recommendation: An Italian red wine such as a reasonably priced nebbiolo from either the Piedmont or Lombardy region has plenty of acidity and body to stand up to the rich taste of this dish.