Prepare the Potato Topping: Add the peeled, chopped potatoes to a large pot of salted water. Bring to a boil and cook until very tender (about 20 minutes). Drain the potatoes thoroughly, then return them to the pot. Set over low heat for a minute or two to help evaporate excess moisture. Pass the hot potatoes through a ricer or mash thoroughly until smooth and lump-free. Mix in the butter, egg yolks, and cream (if using), then fold in approximately two to three tablespoons of grated Parmesan cheese. Season with salt and black pepper to taste. Set aside.

Cook the Lamb Filling: Preheat your oven to 350°F (175°C). Heat olive oil in a large skillet or sauté pan over medium-high heat. Add the ground lamb and cook, breaking apart with a spatula, until beginning to brown and crisp around the edges (about 4–6 minutes). Season with salt and pepper. Add the grated onion, carrot, and garlic. Cook another 2–4 minutes, stirring, until the vegetables have softened and melded into the meat. Pour in the Worcestershire sauce and tomato puree, stirring until well combined. Deglaze the pan with red wine, scraping up any browned bits and reducing the liquid almost completely (about 3–4 minutes). Stir in the chicken stock, chopped thyme, and rosemary. Bring to a simmer and cook until the sauce thickens (an additional 5–7 minutes). Taste and adjust seasoning as needed.

Assemble and Bake: Spread the savory lamb mixture in an even layer on the bottom of an ovenproof casserole or baking dish. Carefully dollop spoonfuls of the mashed potato topping over the lamb, then use a spatula or the back of a spoon to spread and seal the edges. Make sure all the filling is covered. Sprinkle the remaining Parmesan cheese over the surface of the mashed potatoes. Use a fork to rough up the top—this creates signature peaks and valleys that crisp up beautifully during baking. Bake in the preheated oven for 20 minutes, or until the potato crust is golden and the filling is bubbling around the edges. Let the pie rest for 5–10 minutes before serving to allow the filling to settle and flavors to meld.