Position the oven rack in the center of the oven. Preheat the oven to 375 degrees F.
In a large saute pan, heat 1 tablespoon oil over medium-high heat. Add the eggplant slices, season with salt and cook until golden on both sides. Remove to a plate and repeat with the onion slices.
Rub a small flameproof baking dish with the garlic clove. Layer the eggplant, onions, and tomato slices in rows in the baking dish. Drizzle a little more olive oil on top. Cover the baking dish with aluminum foil and bake until heated through and the tomatoes are soft but still hold their shape, about 20 minutes. Remove the baking dish and set the oven to broil. Sprinkle the tomatoes with the shredded cheese and broil to melt and brown the cheese, about another 2 minutes. Remove from the oven and serve.