Melt the butter with the olive oil in a large skillet over medium heat. Add the sliced onions and cook, stirring often, until they turn deep golden brown and soft, about 20 minutes. Remove half the onions and set them aside for the gravy.

In a mixing bowl, combine the ground beef, breadcrumbs, egg, thyme, salt, pepper, and the remaining caramelized onions. Mix until evenly blended but do not overwork the meat.

Press the meat mixture into a pie dish to form an even layer. Lay the puff pastry sheet over the top, trim any excess, and crimp the edges to seal. Cut a few small slits in the pastry to allow steam to escape.

Bake at 375°F for 35 minutes, or until the pastry is golden and the meat reaches an internal temperature of 160°F.

While the pie bakes, return the skillet with the reserved onions to medium heat. Stir in the flour and cook for one minute. Slowly whisk in the beef broth and Worcestershire sauce. Simmer until the gravy thickens, about 5 minutes.

Let the pie rest for 5 minutes after baking, then slice and serve with the warm onion gravy.