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Peel a broad strip of zest from the lemon and then juice the lemon. Put 1 tbsp juice, the zest, sugar, vanilla extract and 500ml water into a saucepan or sauté pan large enough to hold all the fruit in a single layer. Heat gently, stirring a little to help the sugar dissolve. Simmer for 4 mins then remove from the heat.
Peel, halve and core the pears. Add them to a bowl with the remaining juice from the lemon. Heat the syrup again and gently poach the fruit, covered, for 15-20 mins until just tender. Test the pears with the tip of a sharp knife and remove the m with a slotted spoon as soon as they are ready. Lay them in a broad, flat container in a single layer. Allow the poaching syrup to cool slightly (for about 10 mins) then pour over the pears. Cover, then set aside in the fridge to cool completely.
For the frangipane, beat the butter and sugar until light and fluffy. Gradually add half the egg (keep the rest for brushing the pastry), beating well after each addition. Stir in the nuts and the flour.
Roll out 200g of the pastry to make a rectangle measuring 32cm x 14cm. Transfer to a metal baking sheet lightly brushed with a little oil and prick it all over with a fork. Spread the frangipane over the pastry leaving a 2cm border all around it. Lift the pears out of the syrup and pat dry with kitchen paper. Lay these horizontally in alternating directions along the pastry. Brush the borders with water.
Roll out the rest of the pastry into a rectangle measuring 32cm x 17cm. Flour it lightly. Gently fold this over, lengthways, without pressing down. Make cuts horizontally through the fold at 4mm intervals, leaving a 2cm border around the open edge (as if you’re making a paper lantern). Lay this on top of the pears, and unfold it to cover them. Lightly press together the pastry edges and put in your fridge for 30 mins. Heat oven to 200C/180C fan/gas 6. Strain the poaching syrup and boil to reduce it by half.
Trim off about 3mm of pastry all the way round to make the dartois neat. Mix the remaining beaten egg with the milk and use this to brush the pastry. Using the tip of a small sharp knife, make little diagonal markings all along the border. Bake for 25 -30 mins. Brush the tart with the reduced poaching syrup while it’s still warm. Serve with crème fra îche or whipped cream. You can stir 2 tbsp of the poaching syrup into the whipped cream, if you like.