In a large pot, heat 2 tablespoons of olive oil over medium heat. Add 1 diced onion and cook for 8-10 minutes until softened and caramelized.

Increase the heat to medium-high and add 3 cloves of minced garlic, stirring for about 1 minute until fragrant. Then add 1 pound of ground beef, breaking apart with a spatula. Cook for 5-7 minutes until browned; drain excess fat.

Sprinkle in 1 teaspoon of dried thyme, 1 teaspoon of dried rosemary, and season with salt and black pepper. Cook for an additional minute.

Pour in 4 cups of beef broth, using a wooden spoon to scrape the bottom of the pot. Bring to a gentle simmer, stirring occasionally.

Add 2 tablespoons of Worcestershire sauce and 1 bay leaf. Reduce heat to low, cover, and simmer for 1-2 hours.

After simmering, remove the bay leaf and strain the broth. Adjust seasoning if needed, then shred the beef and return it to the pot.

Simmer the shredded beef in the au jus for another 15-20 minutes over low heat.

Preheat the broiler. Lightly butter the crusty rolls and toast them under the broiler for 2-3 minutes.

Assemble the sandwiches by piling shredded beef onto each roll and topping with cheese. Broil for another 2-3 minutes until the cheese is melted.

Serve the sandwiches immediately with warm au jus for dipping.