Preheat your oven to 400°F (200°C) and lightly grease a 9-inch pie dish. Keep your thawed puff pastry in the refrigerator until you are ready to use it so it stays cold.

Heat the olive oil in a large skillet over medium-high heat. Add the minced beef and cook until deeply browned and crumbled, which should take about 5 to 7 minutes. Drain any excess fat from the pan if necessary.

Lower the heat to medium and stir in the chopped onion, carrots, and minced garlic. Sauté the vegetables with the beef for about 5 minutes, or until the onions become soft and translucent.

Sprinkle the flour evenly over the beef and vegetable mixture, stirring constantly for 1 minute to cook off the raw flour taste. Stir in the tomato paste, Worcestershire sauce, and dried thyme until the ingredients are well integrated.

Gradually pour the beef broth into the skillet while scraping the bottom of the pan to release any flavorful browned bits. Bring the mixture to a gentle simmer and let it thicken for 5 to 8 minutes until you have a rich, glossy gravy. Season generously with salt and black pepper, then remove from the heat.

Pour the warm beef filling into your prepared pie dish, spreading it out into an even layer. Roll out the cold puff pastry and drape it over the dish, trimming any excess edges and crimping the sides firmly with a fork to seal.

Cut a few small slits in the center of the pastry to allow steam to escape during baking. Brush the top of the pastry evenly with the beaten egg wash to ensure a shiny, golden-brown crust.

Bake in the preheated oven for 25 to 30 minutes, or until the pastry is puffed, crispy, and deeply golden. Let the pie rest for 10 minutes before slicing to allow the gravy to set, and serve warm.