For the sauce
To prepare the caramel, add a small amount of the sugar to a medium-hot pan set over medium heat and allow it to melt, then add the remaining sugar in a small increments, allowing each addition to fully melt before adding the next. Continue this process until all the sugar has been added to the pan. Cook until a caramel with the desired color develops. Add the apple juice and stir over medium heat to combine. Add the cinnamon stick and apple slices and simmer for 20 minutes. Stir in the butter and brandy and then strain. The sauce is ready to be used or served, or it may be covered and stored in the refrigerator for later use.