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Heat oven to 200C/180C fan/gas 6. Put the shallots into a heatproof bowl and pour over boiling water to cover. Leave for 10 mins, then drain and peel.
Heat the oil and butter in a frying pan. Add the shallots and fry gently for 10-15 mins until softened and lightly browned. Stir in the vinegar and sugar with leaves from 2 thyme sprigs for a few mins until caramelised. Turn off heat and season.
Tear remaining thyme sprigs into a few pieces and scatter over the base of a 22cm ovenproof shallow pan, cake or tart tin without a loose base. Tip in the shallots with all their sticky juices. Roll out the pastry until it’s big enough to cut out a roughly 26cm circle. Lift the pastry circle onto the shallots, then tuck the edges down the inside of the pan. Bake for 25-30 mins until pastry is golden.
Leave tart for 5 mins to settle, then turn out of the tin. Dot with rounds of goat’s cheese and slice into wedges.
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