In a medium bowl, combine the ½ cup brown sugar and ½ teaspoon cinnamon. Add the thinly sliced apples and toss to coat. Set aside.
In a mixing bowl, combine the softened cream cheese, ½ cup white sugar and 1 teaspoon vanilla. Set aside and prepare the puff pastry.
Carefully unfold the sheet of puff pastry. Place on top of a sheet of parchment paper and roll lightly with a floured rolling pin until it is ¼ inch thick. Transfer the puff pastry with the parchment paper to a baking tray.
Add the cream cheese mixture down the center of the puff pastry only.
Arrange the apple slices on top of the cream cheese mixture.
Using a sharp knife or a pizza cutter, begin slicing diagonal strips down both sides of the puff pastry being careful to leave a ½ inch border around the cream cheese and apple slices. You should get around 8 to 10 strips depending on the length of your pastry braid.
Starting at one end, begin folding the strips over the apples, alternating sides to "braid" the dough.
Continue to fold the dough strips until the apples are covered. Tuck the end strips under.
Brush the pastry braid with the beaten egg white and sprinkle the top with 1 to 2 tablespoons white sugar.
Bake in a preheated 400 degree oven for 20 to 25 minutes or until golden brown.
Allow to cool slightly before drizzling caramel sauce on top.
Slice and serve.