In 1-cup glass measure, combine sauce ingredients. Microwave on HIGH for 1 1/2 minutes until bubbly; set aside.
Shape ground beef into four 3/4-inch thick patties. Place onion slices on grid over medium, ash-covered coals. Grill onions, uncovered, 5 minutes.
Add patties; continue to grill, uncovered, for 13 to 15 minutes or until onions are tender and burger centers are no longer pink, turning occasionally. Season burgers with salt after turning, if desired.
About 2 minutes before burgers are done, brush generously with Brew Sauce; top with cheese. Line bottom half of each roll with lettuce. Top each burger with grilled onion and Brew Sauce. Close sandwiches.