Preheat the oven to 375 degrees F. Line a 9-by-13-inch baking dish with parchment.
Heat a grill pan over medium-high heat. Sprinkle the eggplant with some of the salt and pepper.
Combine the olive oil and balsamic vinegar in a small bowl and brush half of the mixture on one side of the eggplant.
Working in batches if necessary, arrange the eggplant brushed-side down in a single layer and cook until you get nice grill marks on the bottom, about 2 minutes. During this time, brush the top of the eggplant with the remaining vinegar mixture. Flip and repeat the grilling on the other side, about 2 minutes. Set aside.
Arrange 2 of the eggplant slices in the prepared baking dish. Top with 1/4 each of the tomatoes and oregano; season with some of the salt and pepper. Top with 1 of the chicken breasts, then sprinkle with 1/4 each of the mozzarella and Parmesan. Repeat these layers, ending with the Parmesan.
Bake until the cheese is browned and the chicken is cooked through, 20 to 30 minutes. Let stand for 5 minutes before serving.