Melt butter in a large skillet over medium heat. Add the sliced onions and cook, stirring occasionally, for about 15-20 minutes until they are golden brown and caramelized. Add minced garlic and sauté for 1 minute until fragrant.

Increase the heat to medium-high and add ground beef to the skillet. Cook until browned, about 5-7 minutes, breaking it apart with a wooden spoon. Drain excess fat if needed.

Stir in dried thyme, Worcestershire sauce, tomato paste, and beef broth. Bring the mixture to a boil, then reduce heat and let it simmer for about 5-10 minutes until thickened and bubbly.

Remove the skillet from heat and stir in half of the Gruyère cheese until fully melted and creamy.

Spoon the beef mixture generously onto toasted buns and top with the remaining Gruyère cheese. Optionally, pop them under the broiler for a couple of minutes to melt the cheese further.