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Cook the potatoes: Boil diced potatoes in salted water until tender (about 10-15 minutes). Drain and mash with a bit of butter, salt, and pepper. Set aside.
Prepare the meat filling: In a large skillet, heat olive oil or butter. Add chopped onion and garlic, sauté until translucent.
Add ground meat and cook until browned, breaking it up as it cooks.
Stir in carrots and peas, cooking for a few minutes to soften.
Mix in tomato paste, Worcestershire sauce, salt, and pepper. Pour in beef broth and simmer until the mixture thickens (about 10 minutes). Adjust seasoning as needed.
Assemble the pie: Preheat oven to 375°F (190°C). Spread the meat and vegetable mixture in a baking dish. Carefully layer mashed potatoes on top, smoothing with a spatula.
Cover the top with puff pastry or roll out the shortcrust pastry and lay it over the pie. Trim edges and press to seal. Brush the top with beaten egg for a golden crust.
Bake: Place the pie in the oven and bake for 25-30 minutes or until the crust is golden brown and crispy.
Let the pie cool for a few minutes before serving.
Serving Tips:
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