Preheat oven to 400°F (200°C). Heat a large skillet or sauté pan over medium-high heat and add olive oil. When oil shimmers, add the chopped onion, diced carrots and diced celery. Sauté until softened and starting to brown, about 6–8 minutes. Add minced garlic and cook 30 seconds until fragrant.

Brown the meat

Add the ground lamb (or beef) to the pan. Break it up with a spatula and cook until no longer pink and nicely browned, about 6–8 minutes. Tilt the pan and spoon off excess fat if there is a large amount, leaving enough to coat the pan.

Build the sauce

Stir in the tomato paste and cook 1–2 minutes to deepen flavor. Sprinkle the flour over the mixture and stir for 1 minute to cook the raw flour taste. Gradually add the beef or lamb stock while stirring to create a slightly thickened sauce. Add Worcestershire sauce, thyme, salt and black pepper. Reduce heat to medium-low and simmer gently 5 minutes until the mixture is cohesive and thickened.

Finish filling

Stir in the frozen peas, cook 1–2 minutes to heat through. Taste and adjust seasoning. Remove from heat and transfer the meat filling to a 9x9-inch (or similar) ovenproof dish, spreading it into an even layer. Wipe out the pan to use for potatoes if desired.

Mashed potato topping

Cook potatoes

Place the peeled and cubed potatoes in a large pot and cover with cold water by about 1 inch. Add a pinch of salt. Bring to a boil over high heat, then reduce to a simmer and cook until potatoes are fork-tender, about 12–15 minutes.

Mash potatoes

Drain potatoes thoroughly and return them to the hot pot for 1 minute to evaporate residual moisture. Add butter, warmed milk, and sour cream (if using). Mash with a potato masher or ricer until smooth but still fluffy. Season with 1 tsp salt and 1/4 tsp black pepper. If using egg yolk for a richer, browner top, temper the yolk with 1–2 tablespoons hot mashed potato, then stir it into the mash.

Top the filling

Spoon or pipe the mashed potatoes over the meat filling, starting at the edges to seal in the filling and working to the center. Use a spatula or fork to create peaks or ridges that will brown in the oven. If desired, sprinkle grated cheddar or Parmesan evenly over the top.

Finish and serve

Bake the assembled shepherd's pie in the preheated 400°F (200°C) oven until the topping is golden and the filling is bubbling at the edges, 20–25 minutes. If you want extra browning, place under the broiler for 2–3 minutes—watch carefully to avoid burning.

Remove from oven and let rest 5–10 minutes. Garnish with chopped fresh parsley, then cut into portions and serve.