Heat grill to medium. On a rimmed baking sheet, place the eggplant and tomato slices in a single layer.
Brush both sides of the vegetables with 4 tablespoons of the oil. Season with ¾ teaspoon salt and ½ teaspoon 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 tablespoon oil and season with ¼ teaspoon salt and ½ teaspoon pepper. Grill until cooked through, about 6 minutes per side.
Transfer chicken to a cutting board and slice each breast into quarters. Spoon the eggplant and tomatoes over the chicken and top with the Parmesan.