In 10-inch nonstick skillet, heat oil over medium-low heat until hot. Cook onion in oil 20 to 25 minutes, stirring occasionally, until brown and softened.
Heat gas or charcoal grill.
Place bratwurst on grill over medium heat. Cook uncovered 10 to 15 minutes, turning frequently, until hot.
Wrap each bratwurst with slice of cheese; place on buns. Top with sauerkraut, onion and mustard.