Heat olive oil in a small saucepan over medium heat. Add the thinly sliced yellow onion and cook, stirring occasionally, for 8-10 minutes until softened and beginning to caramelize.

Add minced garlic and cook for another minute until fragrant.

Pour in the beef broth, Worcestershire sauce, Dijon mustard, dried thyme, black pepper, and optional red pepper flakes. Bring to a simmer, then reduce heat to low, cover, and let it gently simmer for at least 15 minutes while you prepare the sandwiches. This allows the flavors to meld.

While the au jus is simmering, separate out about 2 tablespoons of the caramelized onions from the saucepan for the sandwiches; set aside. Keep the remaining au jus simmering.

Butter one side of each slice of sourdough bread generously with softened butter.

Lay two slices of bread, butter-side down, on a clean surface.

On each of these slices, layer one slice of provolone cheese, half of the roast beef, the reserved caramelized onions, another slice of provolone, and optionally, a slice of Swiss cheese.

Top with the remaining two slices of bread, butter-side up.

Preheat a large skillet or griddle over medium-low heat.

Carefully place the assembled sandwiches into the preheated skillet. Cook for 5-7 minutes per side, or until the bread is golden brown and crispy, and the cheese is thoroughly melted and gooey. You may need to adjust the heat to prevent burning while ensuring the cheese melts.

Once cooked, remove the sandwiches from the skillet and let them rest on a cutting board for a minute.

Slice each sandwich in half diagonally.

Ladle the hot au jus into small individual dipping bowls.

Serve the French Dip Grilled Cheese immediately with the au jus for dipping.