To make the polenta
Heat the oven to 350°F. Coat an 8x8-inch baking dish with the butter. In a medium saucepan, bring the water and the 1 tsp. salt to a boil and slowly whisk in the cornmeal until the mixture is smooth. Reduce the heat to a simmer and continue stirring with a wooden spoon until the mixture is thick, about 10 min. Take the pan off the heat and stir in the thyme and Gruyère. Pour the polenta into the baking dish and bake for 45 min., stirring every 10 min. or so. Meanwhile, make the chicken. (If the polenta is done before the chicken, cover with foil and stir occasionally to keep a crust from forming.)
To make the chicken
Heat the olive oil in a large sauté pan over medium-high heat, but don't let it smoke. Add as many of the chicken pieces, skin side down, as will fit without crowding and brown all sides well, 3 to 4 min. per side. Transfer to a plate, add more olive oil if needed, and brown the remaining chicken. Transfer to the plate. Reduce the heat to medium, add the onion, and sauté until translucent, about 4 min. Add the garlic and cook another 2 min. If the pan is dry, add 1 Tbs. oil. Add the mushrooms and cook until they release their juices and begin to brown, about 5 min. Add the salt and pepper. Add the wine and scrape up any browned bits in the pan. Return the chicken pieces to the pan (they should just fit in snugly) and simmer until the wine is reduced by half, about 5 min. Add the tomatoes and olives and reduce the heat. Simmer the chicken, turning occasionally, until it's completely cooked and very tender, 30 to 40 min. Check the pan frequently and add broth as needed if the pan becomes too dry. Add 1 tsp. of the thyme to the pan.
Serve the chicken over the polenta, with some of the sauce and the remaining thyme sprinkled on top.
nutrition information (per serving):
per serving with skin;
Photo: Scott Phillips