Day before you plan to make the Tarte Tatin peel, core and cut apples into quarters. Half one apple to use in the middle (quarter the other half). Store in fridge until ready to use. (Or use fridge-stored apples) 8 Apples is better for 12 inch cast iron skillet
Preheat oven to 375°F.
Spread butter on bottom and sides of 10 or 12 inch cast iron skillet (or any other oven-safe pan).
Sprinkle sugars evenly over bottom of pan.
Arrange apples on bottom of pan. First put a circular cut piece in middle then arrange rest of the slices around in a circle. Place them upright and close to one another so they can lean on each other for support. Keep arranging until at the edge of pan.
Allow puff pastry to thaw until pliable and roll out onto flour lined surface until 1/8 inches thick. Dust off excess flour and cut into circle big enough to cover apples.
Place puff pastry over apples and tuck into sides of apples, pushing apples together.
On the stove on medium heat start to caramelize the sugars. Allow to cook for about 3 minutes until golden-brown juices begin to bubble around the edges. If juices rise too much, spoon out excess. Continue cooking until the juices are darker brown and smell caramelized, no more than 10 minutes. Transfer to oven and bake for 45-50 minutes until pastry is golden brown and firm.
Take out of the oven and let it cool at least 5 minutes until cool enough to flip. (If there is too much liquid at this point it can be spooned out and then reduced on the stove. On high heat bring to a simmer and allow it to reduce for 5-10 minutes until thicker.)
To flip, place a plate larger than opening and turn out onto serving dish. If any apples remain on skillet, gently remove with finger or spatula and rearrange on pastry.
Serve immediately in wedges by itself or with ice cream.
Enjoy!