In a large bowl combine beef, horseradish mustard, and 1/4 tsp. each salt and black pepper; mix well. Shape into four 3/4-inch thick patties.
For charcoal grill, grill patties on rack of an uncovered grill directly over medium coals for 14 to 18 minutes or until done (160 degrees F), turning once. Top burgers with cheese during the last 1 minute of grilling. (For gas grill, preheat grill. Reduce heat to medium. Place patties on grill rack over heat. Cover; grill as above.) Serve on buns with barbecue sauce. Add arugula, tomato and/or onions. Makes 4 servings.