Heat a large skillet over medium-high heat. Add the ground beef and cook, breaking it up, until well-browned, about 6–8 minutes. Drain excess fat if there’s a lot (leave a tablespoon for flavor).

Add the diced onion and carrots. Cook until softened, about 5–7 minutes.

Stir in the minced garlic and tomato paste; cook 1 minute until fragrant.

Pour in the beef broth and Worcestershire sauce. Sprinkle in the dried thyme. Stir and bring to a simmer.

Reduce heat and simmer until the liquid thickens to a gravy-like consistency, about 8–10 minutes. Stir in the frozen peas and season with salt and pepper to taste. Remove from heat and let cool slightly.

Place quartered potatoes in a pot, cover with cold water, add a pinch of salt, and bring to a boil. Cook until fork-tender, 12–15 minutes.

Drain and return to the pot. Mash with the milk and butter until smooth and creamy. Season with salt and pepper.

Preheat oven to 400°F (200°C). Spoon the beef filling into a 9-inch pie dish, spreading it evenly.

Dollop the mashed potatoes over the filling and smooth the surface, or leave the filling exposed.

Drape the thawed puff pastry over the top, trim and tuck edges. Brush the pastry with the beaten egg and cut a couple of vents for steam.

Bake in the preheated oven until the pastry is puffed and deep golden, about 20–25 minutes. Let rest 5–10 minutes before serving.