Preheat oven to 400 degrees; with rack in upper third. In a medium saucepan, heat 1 tablespoon oil over medium heat. Add the onion and garlic, and cook, stirring, until soft and lightly golden, about 8 minutes. Add tomatoes and salt, and cook, stirring occasionally, until sauce has thickened, about 30 minutes. Stir in vinegar, oregano, and basil; season with pepper. Remove sauce from heat.
Meanwhile, heat a large cast-iron skillet or grill pan over medium heat. Lightly brush eggplant slices with the remaining tablespoon oil. Working in batches, lay slices in skillet in a single layer; cook until browned and they begin to soften, 2 to 3 minutes per side. Transfer to a plate.
Spoon about 1/2 cup tomato sauce into a 9-inch square baking dish, spreading to coat evenly. Arrange eggplant slices snugly in a single layer. Spoon about 1 cup tomato sauce over eggplant, and arrange polenta rounds in slightly overlapping slices on top. Repeat with sauce and another layer of eggplant. Finish by dotting with remaining tomato sauce.
Cover with foil; bake until bubbling and juicy, about 30 minutes. Remove foil; continue baking until sauce is lightly caramelized and eggplant is tender, about 15 minutes more. Remove from oven; let cool slightly, and serve.