In a medium mixing bowl combine beaten egg, bread crumbs, cilantro, onion, 1/4 cup of the chutney, 2 teaspoons of the ginger, cumin, salt, and pepper. Add ground beef; mix well. Shape meat mixture into four 3/4-inch-thick patties.
Grill patties on the rack of an uncovered grill directly over medium coals for 14 to 18 minutes or until done (160 degree F), turning once halfway through. Top with cheese slices. Grill, uncovered, about 2 minutes more or until cheese begins to melt.
In a small bowl stir together the remaining chutney and remaining ginger. Spread chutney mixture onto buns. Serve patties on buns. Serve with sliced onion, sliced tomato, and lettuce leaves, if desired. Makes 4 servings.