Prep the potatoes. Peel and cut potatoes into chunks. Place in a pot, cover with cold, salted water, and bring to a boil. Simmer until fork-tender, about 15–20 minutes.

Make the mashed topping. Drain potatoes well. Mash with butter and milk until smooth. Season with salt and pepper. The mixture should be spreadable but not runny. Set aside.

Heat the oven. Preheat to 400°F (200°C). Lightly grease a 9x13-inch baking dish or a large oven-safe skillet.

Sauté aromatics. In a large skillet, heat olive oil over medium heat. Add onion and celery (if using). Cook 5–6 minutes until softened. Stir in garlic for 30 seconds.

Brown the meat. Add ground lamb or beef. Season with salt and pepper. Cook, breaking it up with a spoon, until browned and no longer pink. Drain excess fat if needed.

Add veggies and flavor. Stir in peas, carrots, and mushrooms (if using). Add tomato paste, thyme, rosemary, and smoked paprika (if using). Cook 2–3 minutes to toast the spices and concentrate the tomato paste.

Build the sauce. Whisk cornstarch into the broth until smooth. Pour into the skillet with Worcestershire and tamari (if using). Add bay leaf. Simmer 3–5 minutes until thick and glossy. Taste and adjust seasoning.

Assemble. Remove bay leaf. Spread the meat mixture evenly in the baking dish. Dollop mashed potatoes on top, then gently spread to the edges to seal in the filling. Use a fork to create ridges for extra browning.

Bake. Place on a sheet pan to catch drips. Bake 20–25 minutes until the top is lightly golden and the edges bubble. For deeper color, broil 1–2 minutes at the end—watch closely.

Rest and serve. Let the pie rest 10 minutes to set. Sprinkle with chopped parsley or chives. Serve warm.