To start, combine the apple juice, sugar, and vanilla extract to a boil in a large saucepan. Stir the mixture continuously so that the sugar melts.
Once boiling, lower the heat to medium and allow the mixture to simmer. It should reduce until it only makes about 1/2 cup of liquid.
Then, stir in the butter until it melts away completely. Finally, add the thinly sliced apple pieces.
Allow them to cook over medium heat until they begin to caramelize and have gone soft. Finally, add the Calvados and give the mixture a good stir. Again, simmer the mixture over low heat until the sauce has reduced by half.
Once the apples are finished, remove them from the saucepan and place them on a plate to cool.
For the caramel sauce, first, place the sugar and water into a heavy-bottom saucepan. Place the pan over medium-high heat and bring it to a boil.
While it heats, continuously brush the sides of the pot with hot water. The hot water will melt the crystals forming on the sides.
Once it starts boiling, lower the heat and allow to color to slowly develop. Once the temperature of the sugar syrup has reached roughly 365-375ºF (185-190.5ºC). Otherwise, wait until it has reached a very deep dark caramel or amber color.
While the sugar syrup is developing, start heating the heavy cream over low-medium heat. Once heated, remove and keep warm until needed.
Once your sugar syrup has reached the right temperature, slowly add the heated cream while whisking constantly. Once you have a uniform mixture, remove it from the heat and add the Calvados and vanilla extract.
Allow the sauce to cool slightly, but not completely. it will become too hard if cooled.
First, pour your caramel sauce over the cheesecake. Make sure to reserve some. Then, top the cheesecake with your homemade apple filling.
And finally, drizzle the apples with the remaining caramel sauce. Serve the cheesecake immediately.