Caramelize the Onions:
In a large skillet over medium heat, melt the butter with the olive oil. Add sliced onions and salt, stirring occasionally for about 25–30 minutes, until deeply golden and soft. Add sugar to help with caramelization, and finish with balsamic vinegar if using. Set aside.

Prepare Burger Patties:
In a large mixing bowl, combine ground beef, salt, pepper, garlic powder, and Worcestershire sauce. Mix gently and form into 4 equal patties, slightly wider than the buns to allow for shrinkage.

Cook the Burgers:
Heat a grill or skillet over medium-high heat. Cook the patties for 3–4 minutes per side or until desired doneness is reached. In the last minute of cooking, place a slice of Gruyère cheese on each patty and cover the pan to melt the cheese.

Toast the Buns:
Spread butter on the cut sides of the buns and toast them in a skillet until golden brown and crisp.

Make the Onion Mayo (Optional):
In a small bowl, mix mayonnaise, onion powder, Dijon mustard, and garlic powder. Adjust to taste.

Assemble the Burgers:
Spread the onion mayo on the bottom bun. Add the burger patty with melted cheese, then pile high with caramelized onions. Top with the other bun and serve hot.