01 - Heat olive oil and butter in a large skillet over medium heat. Add sliced onions and cook, stirring often, until caramelized and golden brown, 12–15 minutes.
02 - Add garlic and dried thyme; stir for 1 minute until fragrant.
03 - Add ground beef to the skillet. Cook, breaking up with a spoon, until beef is browned and cooked through, about 5–7 minutes. Drain excess fat if necessary.
04 - Stir in Worcestershire sauce and beef broth. Simmer over medium-low heat for 5 minutes, allowing the flavors to meld and the mixture to thicken slightly. Season with salt and pepper.
05 - Toast the hamburger buns lightly if desired.
06 - Spoon the beef and onion mixture onto the bottom half of each bun. Top with a slice of provolone or Swiss cheese. Place the top bun over the cheese; if you prefer the cheese melted, briefly broil open-faced sandwiches until cheese bubbles.
07 - Serve immediately while hot.