Combine beer and steak sauce in 1-cup glass measure. Cover and microwave on HIGH 1 to 1-1/2 minutes or until bubbly.
Lightly shape ground beef into four 3/4-inch thick patties. Place patties in center of grid over medium, ash-covered coals; arrange onion around patties. Grill, uncovered, 13 to 15 minutes to medium (160 degrees 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, tomato, and sauce.