Combine ground beef, minced garlic, salt and pepper in large bowl, mixing lightly but thoroughly. Lightly shape into four 1/2- inch thick patties, shaping to fit bread slices.
Place patties on grid over medium ash-covered coals. Grill, covered, 8 to 10 minutes (over medium heat on preheated gas grill 7 to 9 minutes) until instant-read thermometer inserted horizontally into center registers 160 degrees F, turning occasionally.
Place one lettuce leaf on four of the bread slices, top each with burger. Sprinkle evenly with Parmesan cheese; cover with remaining bread slices.