Place the diced potatoes in a large pot and cover with cold water. Add a pinch of salt.
Bring to a boil over high heat. Reduce the heat and simmer until the potatoes are tender, about 15-20 minutes.
Drain the potatoes and return them to the pot.
Add the butter and milk (or cream). Mash until smooth and creamy. Season with salt and black pepper to taste. Set aside.
Make the Meat Filling:
Heat the olive oil in a large skillet or saucepan over medium heat.
Add the chopped onion and cook until softened, about 5 minutes.
Add the minced garlic and cook for another minute.
Stir in the diced carrots and cook for another 5 minutes.
Add the ground lamb (or beef) to the pan and cook until browned, breaking it up with a spoon as it cooks.
Stir in the tomato paste and cook for another 2 minutes.
Add the Worcestershire sauce, beef broth, dried thyme, dried rosemary, salt, and black pepper. Stir to combine.
If you want a thicker filling, sprinkle in the flour and stir well. Let the mixture simmer for about 10-15 minutes until it thickens and the vegetables are tender. If using peas and corn, stir them in at this point.
Assemble the Pie:
Preheat your oven to 400°F (200°C).
Transfer the meat filling to a baking dish, spreading it out evenly.
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