Preheat the oven to 375 degrees F. Lightly grease a 7 by 11-inch baking dish with olive oil.
Heat 2 tablespoons of the olive oil in a large skillet over medium low heat. Add onions, garlic and 1 teaspoon of the salt. Cook, stirring occasionally, for 5 minutes, until completely softened and cooked but not browned. Stir in the basil and thyme. Spread on the bottom of the prepared baking dish.
Toss the sliced vegetables with the remaining olive oil and salt, and pepper. With the long side of the baking dish facing you, alternate the sliced vegetables in compact upright rows on top of the onion mixture.
Cover the dish with aluminum foil and bake until the vegetables are softened, about 30 to 35 minutes. Uncover and sprinkle with the Parmesan. Continue baking until cheese begins to brown, about 20 minutes or so. Serve hot or at room temperature.
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