Combine the peppers, eggplant, tomatoes, zucchini, onion, olive oil and the salt on a large rimmed baking sheet. Roast until beginning to soften and turn brown, 20-25 minutes. Remove the pan from the oven and stir in the garlic, chili, thyme and balsamic vinegar.
Make the polenta
Bring 4 cups of water to a boil in a heavy-duty sauce pan or small Dutch oven. Stir in 1 teaspoon salt. Gradually sprinkle the polenta into the pan while whisking at the same time. Turn the heat to a very low simmer, cover and continue to cook the polenta for 25-30 minutes, until it's thick, fluffy and begins to pull away from the sides of the pan. Stir occasionally so it doesn’t stick to the bottom of the pan. When it's done remove from the heat and stir in the cheese, butter and additional salt to taste, if needed.
Spoon the warm polenta into bowls and put some of the roasted vegetables and the pan juice over each serving. Sprinkle with additional cheese, if you like.
If you don't have a Dutch oven or or enameled cast iron pot, choose your heaviest saucepan and turn the heat to low for best results with minimal stirring.
Storing leftovers or to make ahead: You can refrigerate the polenta and vegetables up to a week, and reheat for about 20 minutes at 350 (175C) degrees. Stir in a little more liquid to the polenta if it's too thick.