Adjust oven rack to middle position and preheat oven to 350 degrees F. In a large bowl, toss mushrooms with 1 tablespoon oil and season to taste with salt and pepper. Coat eggplant with another tablespoon olive oil and season to taste with salt and pepper. Wrap eggplant with heavy duty aluminum foil. Transfer mushrooms and eggplant to a foil-lined rimmed baking sheet. Scatter thyme over mushrooms. Bake, turning mushrooms and wrapped eggplant occasionally until mushrooms are dark brown and eggplant is completely tender , about 45 minutes. Remove from oven, unwrap eggplant, and set aside to cool.
