Combine ¼ cup brown sugar and 1 tablespoon water in a medium saucepan. Cook over medium heat without stirring for about 10 minutes until deep amber. Whisk in 6 tablespoons of unsalted butter, ⅔ cup heavy cream, 2 teaspoons vanilla extract, and a pinch of salt. Let cool.

In a large mixing bowl, beat 16 ounces of cream cheese with ½ cup brown sugar until smooth. Add 2 teaspoons vanilla extract and slowly pour in ⅔ cup heavy cream. Whip until thick and fold in cooled caramel sauce. Pour into the crust and refrigerate for at least 4 hours.

Melt 2 tablespoons unsalted butter in a medium saucepan and add chopped apples, ¼ cup brown sugar, 1 tablespoon lemon juice, and 1 teaspoon cinnamon. Sauté for about 10 minutes until apples are tender. Mix 2 teaspoons cornstarch with 1 tablespoon water, then stir into apple mixture until thickened.

Once the cheesecake filling is set, spoon the cooled apple compote over it. Drizzle with remaining caramel sauce. Slice and serve chilled or slightly warmed.