Heat olive oil in a large skillet over medium-low heat. Add the sliced onions, salt, and pepper. Cook slowly, stirring every few minutes, for 15 to 20 minutes until deeply golden and jammy. Do not increase the heat or the onions will burn instead of caramelizing.

Add the minced garlic and stir for 1 minute until fragrant.

Increase heat to medium-high. Add the ground beef and break it apart with a wooden spoon. Cook until fully browned, about 5 to 7 minutes. Tilt the pan and drain off any excess grease.

Stir in the tomato paste and cook for 1 minute to deepen the flavor. Pour in the beef broth, Worcestershire sauce, and Dijon mustard. Stir well, scraping up any browned bits from the bottom of the pan.

Bring to a gentle simmer, then reduce heat to low. Cook uncovered for 5 to 10 minutes until the sauce thickens enough to hold on a spoon. If it reduces too quickly, add a splash of beef broth.

Toast the brioche buns cut-side down in a dry skillet or under the broiler until golden. Spoon the beef and onion mixture generously onto the bottom bun, top with a slice of Provolone, close with the top bun, garnish with fresh thyme if desired, and serve immediately.