In a large skillet over medium heat, cook the ground beef until browned. Drain excess fat and set the beef aside. In the same skillet, sauté the onions and garlic until softened. Add the diced potatoes and cook for 5 minutes, stirring occasionally.

Stir in the frozen peas and carrots, then sprinkle the flour over the mixture. Pour in the beef broth and Worcestershire sauce, and add the dried thyme, salt, and pepper. Stir well to combine and let the mixture simmer for 10 minutes, or until the potatoes are tender and the sauce has thickened.

Preheat your oven to 375°F (190°C). Roll out one pie crust and place it in a 9-inch pie dish. Pour the beef and potato filling into the crust. Roll out the second pie crust and place it over the filling. Crimp the edges to seal, and cut a few slits in the top crust to allow steam to escape. Brush the top with the beaten egg for a golden finish.

Bake the pie in the preheated oven for 30-35 minutes, or until the crust is golden brown and the filling is bubbly. Let the pie cool for a few minutes before slicing and serving.