Sauté the Veggies: In a skillet, heat a splash of oil over medium heat. Cook the onion, garlic, carrot, and celery for about 5 minutes until they soften and release their aromas.

Brown the Beef: Add the ground beef to the skillet. Cook, breaking it apart, until it’s browned and no longer pink. This should take about 7–10 minutes.

Simmer the Filling: Stir in the tomato paste, Worcestershire sauce, thyme, black pepper, and beef broth. Let the mixture simmer, stirring occasionally, until it's thickened—around 8–10 minutes. Allow the filling to cool afterward.

Prepare the Pastry: Roll out the thawed puff pastry on a floured surface and cut it into your desired shapes, either circles or rectangles, for the pie bases and tops.

Assemble the Pies: Spoon a generous amount of the cooled filling onto half of your pastry pieces. Top with the other half and use a fork to seal the edges tightly.

Vent the Pies: Cut a small slit in the top of each pie to allow steam to escape during baking.

Brush with Egg Wash: Using a pastry brush, coat the tops of the pies with the beaten egg.

Bake to Perfection: Preheat your oven to 400°F (200°C), and bake the pies for 20–25 minutes or until they turn a beautiful golden brown.