Preheat the oven to 180°C/fan 160°C/ gas mark 4.
Place the flour and icing sugar in a food processor, and pulse briefly to mix. Add the roughly chopped cold butter and pulse until the mixture resembles breadcrumbs.
Add the egg yolk, along with one tablespoon of cold water. Pulse until the dough has almost come together. Tip everything out and bring the pastry together by hand. Shape into a flat disc and wrap in cling film, then chill in the fridge for at least 30 minutes.
On a piece of baking parchment, roll out the pastry into a circle, slightly larger than the tart tin, and about 3mm thick. Don’t use any flour on the baking parchment, just sprinkle lightly on the top of the pastry to stop the rolling pin from sticking.
Lift up the baking parchment and invert over the tart tin. Peel back the baking parchment from the pastry and discard. Carefully lift up the edges of the pastry, and ease it down into the tin. Push and press the pastry into every crease of the tin, leaving a small lip over the sides.
Move the tart tin to the fridge for 20 minutes, or to the freezer for 5 minutes. Use a paring knife to trim the pastry, flush against the top of the tart tin. Prick the base of the tart shell with a fork, and then line the tart with ovenproof baking parchment, and fill with uncooked rice. Bake the tart shell for 20 minutes, then lift out the rice parcel. Bake for another 10 minutes, until the pastry is completely cooked.
Peel, core, and slice the apples. Set aside while you prepare the other ingredients. Place the unsalted butter, castor sugar, cinnamon and nutmeg into a large pan over a medium heat, until all the sugar has melted.
Add the apples slices to the pan, and sweat down until the apples are stained with the spices and have softened - about 10 minutes. Leave the apple slices to cool completely.
To assemble, place the apple slices curved side up, against the inside edge of the tart shell. Overlap the slices as you go, to create a spiral going inwards that resembles a large flower. When you come towards the centre, you may need to trim the apple pieces to finish the spiral pleasingly.
In a small milk pan, combine the double cream, milk, and vanilla bean paste. Set over a low heat, swirling occasionally, until the cream mixture starts to steam
Meanwhile, combine the egg yolks and sugar in a batter jug. Whisk them together until they lighten in colour. Pour a little of the hot cream mixture into the eggs, and whisk vigorously to temper the egg mixture. Continue to add a little at a time, whisking continuously, until all the cream mixture is incorporated.
Now you're ready to pour the custard over your assembled apples in the tart shell. Take your time here, and try to direct the custard down into the cracks, rather than just dumping it all on top. Give the baking sheet a little tap against the worktop to help release any air bubbles.
Return the tart to the oven for about 45 minutes, until the custard is just starting to brown in patches. Cool the tart for at least 20 minutes before transferring to a serving plate. Serve warm, or cold.