Heat grill to medium. Place the eggplant and tomatoes in a single layer on a rimmed baking sheet and brush both sides of the eggplant and tomato slices with 4 tablespoons of the oil. Season with 3/4 teaspoon of the salt and 1/2 teaspoon of the pepper. Grill until tender, about 2 minutes per side for the tomatoes, 5 minutes per side for the eggplant. Transfer to a large bowl and toss gently; the tomato slices will fall apart slightly. Brush the chicken with the remaining oil and season with the remaining salt and pepper. Grill until cooked through, about 6 minutes per side. Transfer to a cutting board and slice each breast into quarters. Spoon the eggplant and tomatoes over the chicken and top with the Parmesan.
Tip: When grilling eggplants or other vegetables, make the slices thin, so the inside becomes tender before the outside turns black.