Pat chicken dry and sprinkle with salt and pepper.
Heat oil in a 12-inch heavy skillet over moderately high heat until hot but not smoking, then brown chicken well, starting with skin sides down and turning over once, 10 to 12 minutes total. Transfer chicken to a plate and pour off all but 1 tablespoon fat from skillet.
Add butter, brown sugar, apples, and shallots to fat in skillet and cook over moderate heat, stirring occasionally, until apples are browned, about 5 minutes.
Add broth, vinegar, and sage and deglaze skillet by boiling, stirring and scraping up any brown bits, 1 minute.
Return chicken, skin sides up, to skillet along with any juices accumulated on plate. Reduce heat and simmer, loosely covered with foil, until chicken is cooked through and sauce is slightly reduced, 20 to 25 minutes.