Melt butter in a skillet over medium heat. Add sliced onions, sprinkle with sugar, and cook, stirring occasionally, until onions are golden brown and caramelized, about 20-30 minutes.
Deglaze the pan with balsamic vinegar and season with salt and pepper. Set aside.
Mix the onion soup mix into the ground beef and shape the ground beef into 4 equal patties. Season with additional salt and pepper if desired.
Cook on a preheated grill or skillet over medium-high heat for about 4-5 minutes per side or until desired doneness. Place a slice of Swiss cheese on each patty during the last minute of cooking to melt.
Toast the hamburger buns lightly. Place a cheese-topped patty on each bun bottom, add a generous amount of caramelized onions, and cover with the bun tops. Serve immediately.