Preheat the oven to 400°F. Brown the ground beef in a large skillet over medium heat, breaking it up as it cooks, then drain excess fat.
Add the onion, garlic, carrots, and mushrooms to the skillet. Cook for 5 to 7 minutes until the vegetables soften.
Sprinkle the flour over the mixture and stir for one minute. Slowly pour in the beef broth while stirring, then add the Worcestershire sauce, thyme, salt, and pepper.
Simmer the filling for 3 to 4 minutes until it thickens, then stir in the frozen peas and remove the skillet from the heat.
Transfer the filling to a 9-inch baking dish. Lay the puff pastry over the top, trim any overhang, and cut a few slits in the center. Brush the pastry with the beaten egg.
Bake for 20 to 25 minutes until the crust turns golden brown and the filling bubbles at the edges.