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1
Roll out the pastry into a rectangle and spread the butter over it, leaving a 2.5 cm border.
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2
Sprinkle the Parmesan on top. Fold the bottom third of the pastry up to cover the middle and fold the top third down. Seal the edges, give a quarter turn and roll out a rectangle, then fold as before. Chill for 30 minutes.
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3
Melt the butter in a 23-25 cm round heavy pan (with a lid) that is also suitable for placing in the oven. Add the shallots and garlic, and cook until lightly browned all over.
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4
Scatter the sugar over the top of the shallots and garlic, slightly increase the heat, cook until the sugar begins to caramelize, then turn the shallots and garlic in the pan.
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5
Add the vinegar, water, thyme and seasonings.
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6
Cook, with the pan partly-covered, for 6-8 minutes, until the garlic cloves are just tender.
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7
Remove the pan from the heat and allow the pan and its contents to cool.
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8
Preheat the oven to 190u00b0C (375u00b0F).
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9
Roll out the pastry to the diameter of the pan and lay it over the cooled shallots and garlic.
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10
Prick the pastry with a sharp knife, then bake for 25-35 minutes, until the pastry has risen and is golden.
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11
Set the pan aside for 5-10 minutes to cool, then invert the tart onto a serving platter.
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12
Scatter the tart with a few thyme sprigs and serve.
