In a large skillet with a lid, heat the oil, three turns of the pan, over medium-high. Add the zucchini, eggplant, pepper, onion, garlic and thyme; season with salt and pepper. Cover and cook, stirring occasionally, until the vegetables soften, 5 to 6 minutes. Add the tomatoes and basil. Partially cover and cook, stirring occasionally, until the ratatouille is tender, 10 to 12 minutes more.
In a large saucepan, bring the stock to a boil over medium-high. Gradually whisk in the polenta, add the corn and season. Reduce the heat to medium and cook, whisking constantly, until the polenta is thick, about 3 minutes. Stir in the cheese and butter.
Divide the polenta among 4 bowls. Make a well in the center of each; fill with the ratatouille.