1. Slice the onions and mince the garlic so everything is ready before you start cooking.

2. In a large skillet, melt the butter with the olive oil over medium heat, then add the sliced onions, sugar, a pinch of salt, and some black pepper.

3. Cook the onions, stirring occasionally, for 20 to 25 minutes until they are soft, deeply golden, and caramelized, splashing in a little water or broth if the pan gets too dry.

4. Push the onions to one side of the skillet and add the ground beef to the empty side, breaking it up with a spoon as it cooks.

5. Brown the beef until cooked through, then stir it together with the caramelized onions so everything is evenly combined.

6. Add the minced garlic, Worcestershire sauce, Dijon mustard, and tomato paste, stirring and cooking for about 1 minute until fragrant.

7. Sprinkle the flour over the beef and onion mixture and stir well to coat, cooking for another minute to remove the raw flour taste.

8. Slowly pour in the beef broth while stirring, then add the thyme and bring the mixture to a gentle simmer.

9. Simmer for 5 to 7 minutes, stirring occasionally, until the sauce thickens into a glossy, spoonable consistency that clings to the beef and onions; taste and adjust with more salt and pepper as needed.

10. While the filling simmers, split the brioche or potato buns and toast them cut side down in a dry skillet or under the broiler until lightly golden.

11. Spoon the hot French onion beef mixture onto the bottom halves of the toasted buns, dividing it evenly.

12. Top each sandwich with a slice of provolone or Gruyere cheese and, if desired, place the open sandwiches under the broiler for 1 to 2 minutes until the cheese is melted and bubbly.

13. Sprinkle the tops with chopped parsley or chives, cap with the top buns, and serve immediately while hot and saucy.