Heat 2 tbsp butter and 1 tbsp olive oil in a large skillet over medium-low heat. Add sliced onions and cook slowly for 30-45 minutes until deeply caramelized and sweet. Remove onions and set aside.

Increase heat to medium-high. Add ground beef to the skillet, breaking it apart. Cook for 7-8 minutes until browned. Drain excess grease if necessary.

Reduce heat to medium. Add minced garlic to the beef and cook for 1 minute. Sprinkle flour over the beef and stir constantly for 2 minutes to cook out the raw flour taste.

Slowly pour in beef broth while stirring. Add Worcestershire sauce, fresh thyme, salt, and pepper. Bring to a gentle simmer.

Stir the caramelized onions back into the beef mixture. Simmer on low for 5-10 minutes until the sauce is thick and glossy.

Butter the cut sides of the buns and toast them in a separate skillet until golden brown.

Remove the beef mixture from heat. Lay cheese slices over the meat in the skillet and cover with a lid for 2-3 minutes until melted.

Scoop the cheesy meat mixture onto the toasted buns, garnish with fresh thyme, and serve immediately.