Lightly score each piece of sausage in several places with a sharp paring knife. Heat 1 tablespoon oil in a 12-inch heavy skillet over moderate heat until hot but not smoking, then brown half of sausage, turning occasionally, until golden, about 4 minutes. Transfer to a bowl and cover to keep warm. Pour off fat from skillet and wipe skillet clean. Add remaining tablespoon oil to skillet and brown remaining sausage in same manner, transferring to bowl as cooked.
Wipe skillet clean again, then cook onions with salt and pepper in butter over moderate heat, stirring occasionally, until golden brown, 15 to 20 minutes. Meanwhile, peel, core, and finely chop apple. Stir apple and broth into onion mixture and simmer briskly, uncovered, stirring occasionally, until apple is tender, 6 to 7 minutes. Add sausage and simmer, stirring, until heated through, about 1 minute.
