Gently roll out the puff pastry slightly to around ⅛in thick. Turn an 8in cast iron skillet upside down and place it on top of the pastry so that it has pastry all underneath it. Cut a circle of pastry just slightly larger than the size of the skillet. Lift off the skillet and set aside. Transfer the round of pastry to a parchment or silicone mat-lined baking sheet or plate and place in the refrigerator until needed later.
Peel the apples and cut into quarters. Remove the stem from each piece. Place the apple pieces in a bowl and squeeze a little lemon juice on each piece as it is ready to help stop them going too brown. Repeat with the rest of the apples.
Preheat the oven to 375F/190C.
Cut the butter into cubes and place the butter and sugar in the skillet you used earlier. Warm them together over a medium heat, stirring almost constantly, so that the butter melts and the sugar dissolves. Don't worry if they seem to separate, they should come together shortly after.
Once the sugar has melted and the mixture is starting to caramelize, add the apple slices (if you can, wipe them dry before adding). The skillet may seem overly full, so you can hold back a couple slices at first and add them as the apples start to shrink a little.
Cook the apples in the caramel mixture, turning regularly so that they both get coated in the mixture and start to soften. Keep cooking for around 10 minutes until they are becoming relatively tender when tested with a knife but not mushy. Remove the apple slices and set them aside on a plate then increase the heat under the skillet to reduce the caramel a little (it will likely have become a bit more liquid with the juices from the apple). Stir it regularly then once the mixture has thickened, after a couple minutes, remove from the heat.
Off the heat, add back the apple slices, the outside rounded side down in the caramel. Note the caramel will be hot so avoid touching it. Try to arrange the apple slices in a circle around the outside, just slightly overlapping the slices. Then, use additional pieces of apple in the middle to fill the inner area.
Remove the pastry from the fridge and carefully place the circle of pastry over the apples as centered as you can. Tuck the extra pieces of pastry in around the edges of the apple down inside the skillet then prick the top of the pastry either in a couple points towards the middle or a few dotted around. This is to help steam escape as it cooks.
Place the skillet into the oven and bake for approximately 30 - 40 minutes until the pastry is golden brown and crisp. Remove from the oven and allow it to cool for around 5 - 10 minutes.
Then slide a knife around the edge to ensure it is loosened. Use a plate that is at least the size of the top of the skillet and place it on top of the skillet, top side down, then carefully pick up the skillet and plate together and, still holding the plate on top, flip the skillet over so that the pastry goes onto the plate with the apples on top. You may have some caramel left in the skillet - scoop it out and drizzle over the apples. Serve warm or at room temperature.