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Wash the leeks in a colander but don’t worry about drying them as you want a bit of water clinging on. Heat the butter in a wide saucepan until sizzling, then add the wet leeks and thyme leaves plus salt and pepper. Sweat the leeks really gently for 20-25 mins until they have practically melted but not coloured, adding a little more butter if needed. Set aside and allow to cool.
Heat oven to 220C/200C fan/gas 7. Roll the pastry out to the thickness of a £1 coin. Cut out 4 saucer-size circles and place on a baking sheet. Spread the leeks over the circles leaving a slight border around the edge. Put a slice of goat’s cheese in the middle of each tart and top with a thyme sprig. Pinch the pastry edge together to slightly encase the leeks, then bake the tarts for 25 mins until puffed up and golden. Serve hot and drizzled with flavoured oil if you have it.