Form two beef patties, season with salt and pepper.
Heat skillet, cook patties for 3–4 minutes per side.
Remove and rest patties on a paper towel.
In the same skillet, melt butter for onions.
Add sliced onions, cook until soft and caramelized.
Mix mayo and mustard in a small bowl.
Spread mix on one side of bread slices.
Place cheese, patty, and onions on one slice.
Top with another bread slice, sauce-side down.
Butter outer sides of sandwich.
Grill sandwich on skillet until golden brown.
Flip and repeat on other side. Serve hot!