Cook the polenta according to the package directions.
Meanwhile, heat the oil in a Dutch oven or large pot over medium-high heat. Season the chicken with ½ teaspoon each salt and pepper. Cook, skin-side down, until golden brown, 8 to 10 minutes. Transfer to a plate (the chicken will not be cooked all the way through); reserve the pot.
Add the onion, garlic, and thyme to the reserved pot. Cook, stirring, until the onion is tender, 4 to 6 minutes. Add the tomatoes and their juices, olives, and capers. Cook, breaking up the tomatoes with a spoon, until the sauce starts to thicken, 3 to 4 minutes. Add the chicken, skin-side up, and simmer until an instant-read thermometer inserted into the thickest thigh registers 165° F, 12 to 14 minutes. Discard the thyme sprigs.
Serve the chicken and sauce over the polenta, topped with the thyme leaves.