1. In a large skillet, melt butter and heat olive oil over medium heat. Add sliced onions. Reduce heat to low and cook for 35-45 minutes, stirring every few minutes, or until the onions are very soft and golden brown in color. Remove onions from the skillet and set aside.
2. In the same skillet, add the ground beef. Season with salt and pepper and brown the meat, breaking apart as it cooks, over medium heat. Add beef stock, Worcestershire and all-purpose flour. Stir to combine. Cook additional minute or two to thicken sauce. Remove the beef from the skillet and set aside.
3. Make the Au Jus. In the same skillet, if there is beef fat, skip the butter. Alternatively, if there is little to no beef fat, or you prefer to use butter instead (use a paper towel to soak up the beef fat if you choose this option), add the butter to the skillet and cook over medium heat until melted. Add flour and whisk to combine. Cook over medium heat for 1 minute, then, while whisking, add in beef broth and Worcestershire sauce. Cook, stirring constantly, until thickened. Remove from heat and set aside.
4. Make the sandwich. Butter one side of the slices of sourdough bread. Melt 1 tablespoon butter in large skillet over medium-low heat. Place one piece of sourdough bread, buttered side down in a skillet. Add 2 slices provolone cheese, beef and caramelized onions. Place slice of bread on top, buttered side facing up. Cook over medium-low heat until golden brown, then flip and cook until golden brown on remaining side. Repeat with remaining sandwiches.
5. Serve sandwiches with warm Au Jus. Enjoy!
