Preheat a grill pan over medium-high heat. Add patties, and cook 5 to 6 minutes per side or until no longer pink.
Meanwhile, cook shallots and jalapeño in hot oil in a skillet over medium-high heat 3 to 4 minutes or until lightly browned. Reduce heat to medium; add preserves and vinegar. Cook 3 to 4 minutes or until thickened.
Serve patties on buns with peach mixture and cheese.
For the side dish, toss together all ingredients in a large bowl.