Heat the oil in a heavy medium saucepan over medium heat. Add the onion and sauté until tender, about 5 minutes. Add the water and bring to a boil. Gradually sprinkle in the cornmeal, whisking constantly. Reduce the heat to low and simmer until the polenta is thick and creamy, stirring frequently, about 25 minutes.
Mix the cheese and 1 tablespoon of the thyme into the polenta. Season to taste with salt and a generous amount of pepper. Spoon a generous amount of the polenta in the center of 2 warmed plates. Top with the braised chicken (or another topping). Sprinkle with additional thyme.
Spoon the remaining polenta onto a medium plate. Spread evenly. Cover and refrigerate to use in the White Bean, Chicken, Mushroom, and Butternut Squash Soup (see recipe below).