1 chicken, cut into eight pieces or, a mixture of chicken breasts and thighs
Kosher salt and freshly ground black pepper
1 medium lemon
1 Tbs. unsalted butter; more as needed
1 Tbs. extra-virgin olive oil
5 medium shallots, peeled and quartered lengthwise
3/4 cup jarred brined olives, rinsed, pitted, and halved
8 fresh sage leaves
6 small fresh or 3 dried bay leaves
2 sprigs fresh thyme, plus 1 tsp. chopped
Position a rack in the center of the oven and heat the oven to 425°F.
Season the chicken generously on all sides with salt and pepper.
Cut the ends off the lemon, stand it on one end, and slice off the peel and the bitter white pith to expose the flesh. Cut the lemon segments from the membranes, letting them drop into a small bowl. Cut each segment crosswise into 4 pieces.
Heat the butter and the oil in a 12-inch ovenproof skillet over medium-high heat. Working in batches if necessary, cook the chicken skin side down until golden-brown, 5 to 6 minutes. Transfer the chicken to a plate.
Pour off all but 2 Tbs. of the fat. Add the shallots, olives, sage, bay leaves, thyme sprigs, and lemon segments, and cook until fragrant, 1 to 2 minutes.
Return the chicken to the pan skin side up and transfer to the oven. Roast until an instant-read thermometer inserted into the thickest part of a thigh registers 165°F, 18 to 20 minutes. Serve, sprinkled with the chopped thyme.