Boil the potatoes: Place peeled, cubed potatoes in a large pot of salted water and bring to a boil. Cook until a fork slides in easily, about 15 minutes, then drain well.

Mash the potatoes: Return the drained potatoes to the pot and mash with 3 tablespoons butter and the milk until smooth and fluffy. Season with salt and pepper, then set aside.

Preheat the oven: Set the oven to 400°F so it's ready once the filling is assembled.

Brown the lamb: Melt the remaining butter in a large skillet over medium-high heat and add the ground lamb. Cook, breaking it up with a spoon, until no pink remains and the edges turn deep brown, about 7 minutes.

Soften the vegetables: Add the onion and carrots to the skillet and cook until the onion turns translucent, about 5 minutes. Stir in the garlic and cook for 30 seconds until fragrant.

Build the gravy: Stir in the tomato paste and cook for 1 minute, then pour in the beef broth and Worcestershire sauce. Simmer uncovered until the gravy thickens and coats the back of a spoon, about 10 minutes.

Add the peas: Stir the frozen peas into the lamb mixture and cook for 2 minutes until heated through. Taste and adjust salt and pepper.

Spread the filling: Transfer the lamb mixture to a 9x13-inch baking dish and spread it into an even layer.

Top with mashed potatoes: Spoon the mashed potatoes over the filling and spread to the edges with a spatula. Drag a fork across the surface to create ridges that crisp in the oven.

Bake until golden: Bake at 400°F for 20 to 25 minutes, until the potato peaks turn golden brown and the filling bubbles at the edges.

Rest before serving: Let the pie rest for 5 minutes before serving so the filling sets slightly and doesn't run when sliced.