Sauté Vegetables: In a skillet, heat oil over medium heat and cook the onion, garlic, carrot, and celery for about 5 minutes until they're soft and fragrant.

Brown the Beef: Add the ground beef to the skillet, stirring and cooking until it’s browned all the way through, about 6–8 minutes.

Combine Ingredients: Stir in the tomato paste, Worcestershire sauce, thyme, black pepper, and beef broth. Simmer until thickened, about 8–10 minutes.

Prepare Pastry: Roll out your puff pastry and cut into circles or rectangles.

Fill the Pastry: Spoon filling onto half of the pastry pieces. Top with the other half and seal the edges.

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

Brush and Bake: Brush with beaten egg and bake in a preheated oven at 400°F (200°C) for 20–25 minutes.

Serve and Enjoy: Let cool slightly before serving.