In a large skillet over medium heat, brown the ground beef until fully cooked. Drain any excess fat, then add the chopped onion and garlic. Cook for 3-4 minutes until the vegetables soften. Stir in the tomato paste and cook for another minute to deepen the flavors.

Add the diced potatoes, frozen vegetables, beef broth, Worcestershire sauce, thyme, paprika, salt, and pepper. Bring to a simmer, then reduce heat to low, cover, and cook for 15-20 minutes until potatoes are tender. The mixture should thicken slightly as it cooks.

Preheat your oven to 375°F (190°C). Line a 9-inch pie dish with one pie crust. Spoon the beef and potato filling into the crust, spreading it evenly. Place the second crust over the filling, crimp the edges to seal, and cut a few slits in the top to allow steam to escape.

Brush the top crust with the beaten egg for a golden finish. Bake for 30-35 minutes, or until the crust is golden brown and the filling is bubbling. Let the pie rest for 10 minutes before slicing to allow the filling to set.