Brown the ground beef in a hot large skillet. Stir it often to break up any large lumps. Drain the excess fat to keep the dish light.

Add the grated onions and carrots to the pan. Cook them until they become very soft and fragrant. Stir in the minced garlic for one more minute.

Pour in the tomato puree and the red wine. Let the liquid bubble and reduce by half. This concentrates the flavors for a better sauce.

Add the beef stock and the Worcestershire sauce now. Toss in the fresh thyme leaves for extra aroma. Simmer until the sauce is thick and glossy.

Boil the peeled potatoes in a large pot. Cook them until they are very fork-tender. Drain the water and let them steam for a moment.

Mash the potatoes with butter and egg yolks. Add a splash of milk for a creamy finish. Season with a little bit of salt and pepper.

Spoon the meat mixture into your baking dish. Spread it out into a very even, flat layer. This ensures every bite has enough meat.

Pipe or spread the mash over the beef. Use a fork to create small peaks and ridges. These ridges will get very crunchy in the oven.

Top the pie with a lot of Parmesan cheese. Bake at 200°C for about twenty to thirty minutes. Remove when the top is golden and bubbling.