1. Preheat grill pan over medium-high heat. Add patties, and cook 5 to 6 minutes per side or until no longer pink. Remove pan from heat. Add cheese to patties; cover until cheese melts.
2. Meanwhile, cook shallots in hot oil in a small saucepan over medium-high heat 3 minutes or until starting to brown.
3. Add preserves, vinegar, rosemary, and salt; reduce heat to medium, and cook 3 to 4 minutes or until mixture is pourable. Spread preserves mixture on buns. Serve burgers on buns with arugula, if desired.
