Caramelize the onions
Heat butter and olive oil in a large skillet over medium heat. Add sliced onions and cook slowly for 18–22 minutes, stirring occasionally, until deep golden brown, soft, and sweet. Add garlic during the last minute and cook until fragrant.

Brown the beef
Push onions to the side of the pan. Add ground beef and cook over medium-high heat, breaking it apart until fully browned with lightly crisp edges. Drain excess grease if needed.

Create the French onion sauce
Sprinkle flour over the beef and onions. Stir and cook for 1 minute. Slowly pour in beef broth while stirring. Add Worcestershire sauce, thyme, salt, and pepper.

Simmer until thick
Reduce heat to medium-low and simmer for 5–7 minutes until the mixture becomes thick, rich, and glossy. The filling should be spoonable, not watery.

Melt the cheese
Lower heat and place cheese slices over the beef mixture. Cover the skillet for 1–2 minutes until fully melted and creamy.

Toast and assemble
Lightly toast buns until golden. Spoon the French Onion Beef Sloppy Joes mixture onto the bottom buns, add tops, and serve immediately