Preheat the oven to 180°C/gas 4.
Heat the oil in a large pan until hot. Season the mince and fry in the oil over moderate to high heat for 2-3 minutes. Stir the onions and carrot into the mince then grate the garlic in as well. Add the Worcestershire sauce, tomato puree and herbs and cook for 1-2 minutes, stirring constantly. Pour in the red wine and reduce until almost completely evaporated. Add the chicken stock, bring to the boil and simmer until the sauce has thickened.
Meanwhile, cook the potatoes in boiling salted water until tender. Drain then return to the hot pan over low heat to dry out briefly. Pass them through a potato ricer then beat in the egg yolks, followed by about 2 tbsp grated Parmesan. Check for seasoning.
Spoon the mince into the bottom of a large ovenproof dish. Using a large spoon, layer the mashed potato generously on top of the mince, starting from the outside and working your way into the middle. Grate some extra Parmesan over and season. Fluff up the mash potato with a fork to make rough peaks. Bake in the oven for approximately 20 minutes, until bubbling and golden brown.
For the carrots bring a pan of water to the boil with the rosemary, thyme and garlic. Boil the carrots in the water for 3 minutes to soften slightly, then drain and pat dry. Heat the oil in a large non-stick sauté pan then add the carrots and some seasoning. Brown the carrots all over until just tender, adding the butter towards the end of cooking.
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