Preheat the oven to 400°. Spread the oil on 2 baking sheets. Arrange the tomato slices in a single layer on 1 sheet and the eggplant slices on the other. Season the vegetables with salt and pepper and sprinkle with the herbs. Bake the vegetables for 30 minutes, then turn the slices and bake for 10 minutes longer, or until the eggplant is golden on both sides and the tomatoes are jamlike. Remove from the oven and reduce the heat to 350°.
Using a spatula, transfer one-third of the tomatoes in an even layer to a 9-inch square baking dish. Cover them with half of the eggplant and one-third of the Monterey Jack. Repeat. Top with the remaining tomatoes and add any tomato juices from the baking sheet. Sprinkle with the remaining Monterey Jack and the Parmesan. Bake for 20 minutes, or until lightly golden. Cover the dish with foil and set aside for about 15 minutes before serving.
The cheeses point to a medium-bodied red as the best accompaniment. Consider a Rioja from Spain, such as Paternina Banda Azul Crianza or Conde de Valdemar Crianza.