Caramelize the Onions:

1. In a large skillet, heat butter and olive oil over medium heat. Add sliced onions, sugar, and salt.
2. Cook, stirring occasionally, for about 20-25 minutes until onions are golden brown and caramelized.
3. Add beef broth and wine (if using) to deglaze the pan, scraping up any browned bits. Cook until the liquid is mostly evaporated. Remove from heat and set aside.

Prepare the Burgers:

1. Preheat your grill or stovetop griddle to medium-high heat.
2. In a large bowl, combine ground beef, Worcestershire sauce, salt, and pepper. Mix gently to combine, being careful not to overwork the meat.
3. Divide the mixture into 4 equal portions and shape into patties.

Cook the Burgers:

1. Grill the patties for about 4-5 minutes per side, or until they reach your desired level of doneness.
2. In the last minute of cooking, place a slice of Gruyère cheese on each patty and allow it to melt.

Assemble the Burgers:

1. Brush the cut sides of the brioche buns with melted butter and toast them on the grill or stovetop until golden brown.
2. Place a cooked patty on the bottom half of each bun, then top with a generous portion of caramelized onions. Sprinkle with fresh thyme leaves if desired. Top with the other half of the bun.

Serve:

1. Serve immediately, with a side of extra beef broth for dipping if you like.