In a large bowl combine egg, bread crumbs, smoked cheddar, green onions, Worcestershire sauce, mustard, and pepper. Add ground beef and mix gently. Firmly shape into four 1/2-inch-thick patties; set aside.

Heat a nonstick or well-seasoned grill pan on stovetop over medium-high heat until hot. Reduce heat to medium; add onion slices, if using, to grill pan. Grill for 6 to 8 minutes or until tender and lightly browned, turning once. Remove from pan.

Add patties to pan. Cook for 9 to 12 minutes or until meat is done (160 degrees F), turning once.

Spread bottoms of hamburger buns with chutney or relish; top with arugula or watercress, patties, onion slices, if using, shaved cheese, and bun top. Makes 4 servings.

Preheat grill according to manufacturer's directions. Add onions, if using, to grill. If using a covered grill, close lid. Grill until onions are lightly browned. For covered grill, allow 5 to 7 minutes. (For open grill, allow about 10 minutes.) Remove onions. Add patties to grill. If using a covered grill, close lid. Grill until meat is done (160 degrees F). For a covered grill, allow 4 to 6 minutes. (For open grill, allow about 7 to 8 minutes per side.) Continue as directed in Step 4 above.

Rinse arugula thoroughly, it can be gritty. Dry in a salad spinner or pat dry with paper towels before laying on burgers.