Preheat oven to 450F with rack in middle. Pat chicken dry and sprinkle with 1 1/4 teaspoons salt and 1/2 teaspoon 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. Strain sauce through a sieve into a bowl. Whisk in mustard, chives, and salt and pepper to taste. Serve chicken with sauce.