Place pork chops, bourbon, garlic, brown sugar, soy sauce, and mustard in a resealable plastic bag. Seal bag; refrigerate at least 8 hours.
Spray grill rack with nonstick nonflammable cooking spray. Preheat grill to high heat (400° to 450°). Preheat a 12-inch cast- iron skillet on grill 15 minutes. Remove pork from marinade; reserve marinade.
Grill peaches, directly on grill rack, until tender, 2 to 3 minutes per side. Set aside.
Place pork chops in skillet; grill, covered, 4 minutes per side. Remove pork from skillet; keep warm. Add shallot to skillet; cook, stirring occasionally, 2 minutes. Add reserved marinade and broth to skillet. Bring to a boil; cook, whisking frequently, until slightly thickened, approximately 5 minutes. Stir in peaches, butter, and lemon juice; cook 1 minute more. Carefully remove skillet from grill. Stir in basil. Serve peach mixture over pork chops.