Heat a large skillet over medium-high heat. Add the ground beef and break it up with a spoon. Cook until browned and no longer pink. Drain excess fat if there’s a lot.

Add the diced onion and carrots to the skillet. Sauté until the onion is translucent and carrots begin to soften, about 6–8 minutes.

Stir in the minced garlic and cook 30–60 seconds, until fragrant.

Mix in the tomato paste and cook for another minute to deepen the flavor.

Pour in the beef broth and add Worcestershire sauce and dried thyme. Stir well.

Simmer gently until the sauce reduces and thickens slightly, about 6–10 minutes. Season with salt and pepper.

Stir in the frozen peas just before removing from heat so they heat through without getting mushy.

While the filling simmers, place the peeled and quartered potatoes in a pot of cold salted water. Bring to a boil and cook until tender, about 15–18 minutes.

Drain well, return to the pot, then mash with butter and milk until smooth and creamy. Season to taste with salt and pepper.

Preheat the oven to 400°F (200°C).

Spread the beef-vegetable filling evenly in a 9×13-inch or similarly sized baking dish. Smooth the mashed potatoes over the filling in an even layer.

Drape the thawed puff pastry sheet over the potatoes. Tuck edges down into the dish or trim to fit. Beat the egg and brush it over the pastry for a glossy, golden finish.

Bake in the preheated oven until the pastry is puffed and deeply golden, about 20–25 minutes. If filling bubbles too vigorously, tent loosely with foil.

Let rest 5–10 minutes before slicing to allow the filling to settle.