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Peel the apples, cut them into quarters and cut out the core using a pairing knife.
Put the caster sugar or maple syrup, olive oil, star anise, cardamom pods and cinnamon stick into a 24 cm / 9.5" ovenproof frying pan / skillet.
Scrape the seeds from the half a vanilla pod and put them and the pod itself into the pan.
Place on a medium heat and give it a good stir. If using sugar, the mixture will look like an olive oil granita until the sugar melts into the oil. After a few minutes of gentle bubbling, the mixture will separate, turn an amber colour and start to bubble more vigorously. Watch the mixture carefully and the minute the sugar turns dark amber (don't push it too far as it will burn), take the pan off the heat.
Add the cut apple pieces into the pan gradually, coat them in the caramel mixture and start to simmer on a low heat. As the apple starts to shrink you should then be able to fit all of the apple pieces in the pan. Caramelise the apples for another 15 minutes until they turn golden. Move the pieces around the pan so that they caramelise at the same pace. The sides touching the pan will cook quicker so if some pieces look more done than others, remove them from the pan and set aside on a small plate so that the other pieces have a chance to catch up.
Add brandy and simmer for another 10 minutes (until the apple pieces are the same colour as the 2nd and 3rd photo). By the end, the apples should be pre-cooked enough that you should be able to insert a skewer into the flesh with a considerable amount of resistance.
Preheat the oven to 200° C / 390° F.
Leave the apple mixture to cool for 15 minutes so that you can handle the apples with your hands. At this point arrange the apple pieces on a side, in a spiral shape. Make sure the cinnamon stick, cardamom pods, vanilla pod and star anise are on the pan itself so they can be removed easily once the tart is cooked.
Roll out the puff pastry to 3 mm / 0.1" and, using a plate that is slightly larger than the pan, cut out a circle of pastry.
Drape the pastry over the pan and make sure you tuck in all the edges down the side of the pan. This will give you that rustic tarte tatin crust.
Pierce the pastry 3-5 times with a skewer to allow the steam to escape and bake in the oven for 15 minutes.
Turn the oven temperature down to 180° C / 355° F and cook for a further 15-25 minutes until the pastry is golden brown. Be careful not to burn yourself when removing the pan out of the oven as the pan's handle will be piping hot.
Leave the tart to cool for at least 15 minutes (making sure the pan handle has cooled from the oven!). Place a plate on top of the pan and, holding the plate against the pan, flip the pan upside down
Serve on its own or with some vanilla ice cream or crème fraîche.