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Heat half the butter and the oil in a large pan, add the onion and carrots, and cook for about 10 mins until soft. Add the game and sausagemeat to the pan. Turn up the heat, squash the meat with the back of your spoon to break it up, and cook for 10 mins until nicely browned.
Stir the flour and tomato purée into the meat, cook for 1 min, then add the wine, stock, sauce, bay and 3 tsp thyme. Bring to a simmer, reduce the heat, cover with a lid and cook for 1 hr. Remove the lid and cook for 15 mins more until the meat is tender and the sauce has thickened.
Put the potatoes and veg in a large pan, cover with water, bring to the boil and cook for 20 mins until really tender. Drain, leave to steam for 5 mins, then mash with seasoning, milk and the remaining butter.
Heat oven to 220C/200C fan/gas 7. Tip the meat into an ovenproof dish and spoon on the mash. Sprinkle with the remaining thyme and bake for 45 mins until the top is golden and the sauce bubbling.