Heat olive oil and butter in a large skillet over medium heat. Add the sliced onions and season with ½ teaspoon salt and ¼ teaspoon black pepper. Cook for 15–20 minutes, stirring occasionally, until soft and deeply golden. Stir in Worcestershire sauce and thyme and cook for 1–2 minutes more. Set aside.

While the onions cook, divide the ground beef into four equal portions and gently shape into patties about ¾-inch thick. Season both sides with the remaining salt and black pepper.

Heat a large skillet or grill over medium-high heat. Cook the burgers for 4–5 minutes per side, or until they reach your desired doneness.

During the last 1–2 minutes of cooking, place a slice of Gruyère on each burger. Cover briefly to allow the cheese to melt.

Toast the buns and spread with Dijon mustard. Place each burger on a bun and top generously with caramelized onions. Add arugula or mixed greens, if desired.

Serve immediately. Enjoy!