Cut onions crosswise into 1/2-inch thick slices, peel, and separate into rings. Discard the inner bulbs.
Heat oil to 365℉ in a deep fryer or heavy-bottomed pot attached with an oil thermometer.
Combine beer with one cup flour; mix thoroughly. Place remaining flour in a shallow dish.
Dredge onion rings in flour, shake off excess, and dip in beer batter. Spiral slices of bacon around each onion ring to completely wrap it (you may need several slices depending on the size of the onion). Dip into batter again.
Deep-fry the rings, a few at a time, for 3-5 minutes, or until deep brown. Transfer to paper towels to drain.
Serve while hot with cheese sauce.