Using a 9-inch (23 cm) flat-sided cake pan as a template, cut a circle out from the puff pastry. Using a fork, poke holes all over to provide ventilation. Set aside.
Peel and quarter the apples, using a spoon or melon baller to remove the cores.
Preheat oven to 375°F (190°C).
In a large saucepan over medium heat, distribute the water and sugar evenly and cook until light amber in color, stirring to help melt any lumps, 5-7 minutes. Add the butter, stirring constantly until the color is a creamy light brown. Add the apples, stirring until they are coated in a thick layer of caramel.
Cook for about 15-20 minutes, turning the apples constantly so that they bathe in the caramel. Remove from the heat when the caramel has reduced and little remains in the bottom of the pan. Be careful not to burn the caramel, tasting it from time to time to ensure it does not taste bitter.
Arrange the apple slices in concentric circles on the bottom of the cake pan. Press the apples tightly against each other, then pour the remaining caramel over the top.
Lay the circle of puff pastry on top. Tuck the puff pastry down the sides of the pan.
Bake for 45-50 minutes, until the pastry is golden brown and firm. Cool for about 1 hour, then invert onto a plate.