There are as many versions of this classic of French patisserie as there are cheeses! For this version, Prue has added calvados to the frangipane and put the filling in a pâte brisée shell, for its crumbly, buttery texture.
Make the pâte brisée. Tip the flour and salt into a bowl. Rub the butter into the flour mixture with your fingertips until it resembles breadcrumbs. Make a well in the middle, add the egg yolk (reserving the white) and 2 tablespoons of the ice-cold water. Using a table knife, bring the mixture together to form a dough, adding the extra 1 tablespoon of water if needed.
Lightly flour the work surface, tip the pastry out of the bowl and shape the dough into a ball, then wrap and chill it for 20 minutes.
Make the apple purée. While the pâte brisée is chilling, tip the apples into a pan, add the lemon juice and butter, then simmer for 20 minutes, until the apples are soft and any liquid has evaporated. Remove from the heat and crush the apples with a potato masher or wooden spoon to a coarse purée. Leave to cool.
Make the pastry case. Roll out the pâte brisée on a lightly floured work surface to a 30cm round and use the round to line the tart tin. Chill for 20 minutes.
Heat the oven to 180°C/160°C fan/350°F/Gas 4. Line the pastry case with baking paper and baking beans and bake blind for 20 minutes. Meanwhile, whisk the reserved egg white with a fork. Remove the baking beans and paper and brush the inside of the case with the egg white. Return the case to the oven and bake for 10 minutes to dry out the base. Leave to cool. Turn the oven off for now.
Make the lemon syrup. Pour 100ml of water into a pan, add the sugar and lemon juice and bring to the boil. Reduce the heat and simmer for 2-3 minutes, until the sugar dissolves. Pour the syrup into a large bowl and leave to cool slightly.
Make the frangipane. Beat the butter and sugar in a stand mixer fitted with the beater, on medium speed for 3-5 minutes, until pale and creamy. Add the egg and mix until combined. Turn the speed to low, add the ground almonds and calvados and mix until just combined; do not overmix.
Spoon the cold apple purée into the pastry ease and level it with a palette knife, then spoon blobs of the frangipane on top and spread it out into an even layer.
Make the topping. Reheat the oven to 180°C/160°C fan/350°F/Gas 4. Place the apple slices in the warm lemon syrup and turn to coat. Working from the outside edge of the pastry case, dab each apple slice on kitchen paper to remove any excess syrup and arrange the slices in concentric overlapping circles. When you reach the middle, slice a few of the apple slices in half so they are wafer thin and shape them into an open rose to fill the hole in the centre of the tart. Brush all the apple slices with the melted butter and bake for 30 minutes.
Make the apricot glaze. Crush the apricots and jam sugar together in a small pan using the potato masher, then place the pan on a low heat. Leave the fruit and sugar to slump and then increase the heat slightly to bring the mixture to the boil. Immediately reduce the heat again to a strong simmer, and cook for 4 minutes, until you have a jammy consistency. Remove from the heat and pass the mixture through a fine-mesh sieve. Stir 2 tablespoons of water into the purée to give a thin glaze.
Remove the tart from the oven, leave it to cool for 5 minutes, then brush the top with the apricot glaze.