Saute your onions (sliced) with stout beer in a skillet on low heat for at least thirty minutes or until soft. Add the desired amount of stout halfway through the cooking process to allow onions to absorb flavor.
Cook your Burgers to order (4-5 minutes per side on high heat for medium-rare). Melt cheese on top of Burgers.
Stack Burgers atop the bottom half of brioche bun, then top with onions. Spread horseradish mayo on the bottom of top bun and place on top.