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Cook the swede and potatoes in a pan of boiling salted water for 10 mins until tender, then drain and set aside.
Heat oven to 200C/180C fan/gas 6. Heat the oil in a large frying pan over a low heat. Add the onion and carrots, and gently fry for 5 mins. Add the mushrooms, increase the heat to medium and cook for a further 5 mins. Tip the vegetables into a bowl.
Add the mince to the frying pan and cook until browned. Pour away any excess fat. Add the vegetables (not the swede and potatoes), tomatoes and tomato purée, cover and simmer for 15 mins, topping up with a little water if the mince looks a bit dry.
When the swede and potatoes are cool enough to handle, coarsely grate; then toss in the melted butter and season.
Tip the mince mixture into a deep ovenproof dish, top with the grated swede and potatoes, then put in the oven and cook for 40 mins.