Combine beer and steak sauce in 1-cup glass measure. Cover and microwave on HIGH 1 to 1-1/2 minutes or until bubbly.
Place patties in center of grill over medium, ash-covered coals; arrange onion around patties. Grill, uncovered, 13 to 15 minutes to medium (160°F) doneness, until patties are not pink in center and juices show no pink color and onions are tender, turning occasionally. Season with salt after turning. About 2 minutes before patties are done, brush generously with sauce; top with cheese.
Serve burgers and onions in rolls with lettuce and sauce.