Step 1: Heat the oil in a Dutch oven or large pot over medium-high heat. Season the chicken with 1/2 teaspoon each salt and pepper. Cook, skin-side down, until golden brown, 8 to 10 minutes.
Step 2: Transfer the chicken to a plate (the chicken will not be cooked all the way through); reserve the pot.
Step 3: 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.
Step 4: Add the chicken, skin-side up, and simmer until an instant-read thermometer inserted into the thickest thigh registers 165° F, about 12 to 14 minutes.
Step 5: Meantime, prepare the polenta according to package directions. You want it creamy, not hard.
Step 6: When the chicken is done, discard the thyme sprigs. Serve the chicken and sauce over the polenta, topped with the thyme leaves
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