1Heat gas or charcoal grill. In 2-quart saucepan, heat cider to boiling. Add bratwurst; reduce heat to low. Cover; simmer 15 minutes.
2Meanwhile, brush oil on cut side of onion. Place onion, oil side up, on grill over medium heat. Cover grill; cook 9 to 12 minutes, turning once, until onion is soft and edges are golden brown. Remove onion from grill; coarsely chop. In small bowl, mix onion, applesauce and mustard; set aside.
3Drain bratwurst. Place on grill over medium heat. Cover grill; cook 6 minutes, turning once, until brown.
4In each bun, place 1 bratwurst and about 2 tablespoons relish.