Place butter in skillet over medium heat until melted. Add onions and cook about 10-15 minutes or until soft, stirring occasionally. Add broth, 1/4 cup beer, brown sugar, 1/4 teaspoon salt, and 1/8 teaspoon pepper; stirring occasionally. Cook another 5-10 minutes or until browned and liquid has reduced.
Meanwhile, in a small bowl, mix together mayonnaise, dijon mustard, and horseradish until well combined, set aside.
Preheat grill to medium-high heat. In a large bowl combine ground beef, 2 tablespoons beer, garlic, 1 teaspoon salt, and 1/2 teaspoon pepper until just combined. Do not overmix. Make into 4 patties. Grill 3-5 minutes per side or until desired doneness. During last minute of grilling place rolls on grill (cut side down) to toast and place cheese on patties to melt.
To assemble, spread some of the sauce on the cut side of the rolls, top bottom part of each roll with a patty, onions, then top part of roll.
SNAPPY TIPS: I used ground chuck for the burgers but any type of ground beef would work well.SNAPPY SUBSTITUTIONS: Instead of beer you could use red wine or beef broth.