Preheat oven to 450°F with a rack in middle. Pat chicken dry and season generously with salt and freshly ground black pepper. Heat oil in an ovenproof 12-inch heavy skillet over medium-high heat until it shimmers. Working in 2 batches, brown chicken, skin side down first and turning once, about 5 minutes per batch. Return all chicken, skin side up, to skillet and roast in oven until just cooked through, 15 to 20 minutes. Transfer chicken to a platter, then add shallots, wine, and broth to pan juices in skillet and boil, scraping up any brown bits, until reduced by half, 2 to 3 minutes. Add cream and boil until slightly thickened, about 1 minute. To thicken the sauce further, turn the heat to high and boil it until it reduces to a consistency you prefer. Strain sauce through a sieve into a bowl, if desired. Whisk in mustard, chives, and salt and pepper to taste. Serve chicken with sauce.