1. Begin by heating olive oil in a large skillet over medium heat. Add the onions and cook until caramelized, about 10-15 minutes, stirring occasionally.
2. Add minced garlic to the onions and sauté for another minute until fragrant.
3. Push the onions to one side of the skillet and add the ground beef. Cook until browned, breaking it up with a spoon, about 5-7 minutes.
4. Stir in Worcestershire sauce, tomato paste, beef broth, and dried thyme. Let the mixture simmer for 10 minutes until slightly thickened. Season with salt and pepper to taste.
5. Meanwhile, preheat your broiler. Split the hamburger buns and place them on a baking sheet. Top each bun with a slice of provolone cheese and broil until the cheese is bubbly and golden, about 2-3 minutes.
6. Spoon the beef mixture onto the bottom half of each bun, top with the cheesy bun, and garnish with fresh parsley if desired.

Remember, the secret to this dish is patience with the onions. Letting them caramelize fully will bring out their natural sweetness and create that deep, rich flavor we’re aiming for.