Add butter and apples to a skillet and heat over medium heat, stirring frequently until butter is melted and apples begin to get tender, about 6-8 minutes.
Add brown sugar and cinnamon and mix well.
Mix together cornstarch and water and pour into skillet and mix, cooking for about a minute or until mixture is thickened. Remove from heat and cool slightly.
Preheat oven to 425 and unroll a puff pastry sheet, keeping the paper on the dough.
Slice the pastry into 8 squares, or rectangles, depending on the shape of your pastry. Spoon apple filling into each square evenly, placing into the bottom corner of the square, leaving the filling about 1/2" from the edge.
Fold the corners together to create a triangle shape. It may be slightly difficult with the pastry shape if it's a rectangle, just fold the dough as best as you can. Also, work quickly, as the pastry begins to come to room temperature it gets very sticky and is challenging to work with. Crimp the edges of the turnover with a fork.
Scramble egg and gently use a pastry brush to add an egg wash to the top of each turnover.
Bake for 15 minutes or until golden brown. The paper will naturally turn dark brown. Remove from oven and let cool.
Mix together powdered sugar and water (or milk) and drizzle glaze on top of turnovers. Let stand to harden before serving.