Pre heat oven to 375 F.
Gently roll out your pastry dough on a floured surface. (Place the pan you will be baking on the puff pastry to measure the circle you will need). You need to cut a circle little bit bigger than the pan about an inch bigger. Cut it into a round that will fit inside the pan leave it aside once you cut the circle. (see notes)
Peel the skin of the apples. Cut the apples as close to core as possible into three quarters; set aside. No need for perfect size. Remove the seeds. Coat them in lemon juice. Leave them aside.
Take a heavy bottom oven proof pan about 9 inch or 10 inch in size (read notes) add sugar to it and let it gently dissolve over medium to low heat. Do not be tempted to stir or use a spatula. You can shake the pan. Cook it until it becomes golden brown. It will take 6-8 minutes for this step. You cannot rush this step and must be very careful not to burn the sugar. Cook it until the sugar looks dark brown or amber color. You can test the color on plain white plate if you are not sure in the pan.
Remove it from the heat, immediately drop the cubes of butter and whisk to melt and combine. This will also cool down the caramel and stop the cooking further.
Bring it back to heat, add cinnamon and cardamom powder, vanilla extract and star anise, vinegar and salt. Arrange the apple quarters cut side down in the bottom of the pan in concentric circles. Cover the pan completely with apples also in the middle of the pan. Pack them tightly. Apples shrink a lot during baking. Let it cook for 10 more minutes over low heat.
Carefully take your puff pastry and top it on the apples and tuck all the sides carefully. Be careful if the pan is hot. Cut few vents or slits with the help of knife on puff pastry. The reason we do this is to let the air vent out while baking and place pan on a baking sheet. Bake for 25 to 30 minutes, or until pastry is nicely browned/golden on top and puffed up.
Once baked, remove it from the oven and let it cool down in the pan for 10 minutes. I like to rest it as the caramel gets some time to thicken up as the pan cools down.
Run a knife around the edges of the baked tarte and to invert, top it with a serving plate and carefully preferably using oven gloves, invert the pan upside down to transfer the baked tarte.
Remove the pan, and if any apples are stuck on the pan don’t worry just take them and arrange them on the tarte. There will be liquid caramel as well. Just let all of it be in the tarte.
Serve it warm topped with crème fraiche, if desired
Store leftovers in the fridge and just rewarm it before serving.