Preparation

• Position a rack in the center of the oven and heat the oven to 450°F.
• If portions of the backbone are still attached to the chicken quarters, cut them off and discard. Pat the chicken dry with paper towels. In a small bowl, combine the olive oil, thyme, salt, and red pepper flakes. Using your fingers, rub all of the seasoned oil over all the chicken pieces, carefully separating the skin from the meat and rubbing the oil under the skin as well. Arrange the legs skin side up (not overlapping) on a heavy-duty rimmed baking sheet. Roast the chicken for 20 minutes.
• Meanwhile, put the lemon slices and olives in a small bowl. When the chicken has roasted for 20 minutes, take the pan out of the oven and spoon a little of the fat from the pan over the lemons and olives (or use a little fresh olive oil). Scatter the lemons and olives on the baking sheet around but not on top of the chicken, trying to keep the lemon slices away from the edge of the pan where they might burn.
• Continue to roast until the chicken juices run clear when pierced with a knife and an instant-read thermometer inserted in a thigh registers 170°F, another 20 to 25 minutes. Transfer the chicken, olives, and lemon slices to a platter and serve.