Melt the butter in a saucepan over medium heat. Add the brown sugar, cinnamon and apples and cook until the apples are caramelized and have started to soften, 10 to 15 minutes. Transfer to a bowl and let cool, then keep in the refrigerator.
Preheat the oven to 400 degrees F (200 degrees C).
For the turnover:
Roll out the puff pastry to 10 by 10-inch square with a 1/8-inch (3mm) thickness on a lightly floured surface. Transfer to a large baking sheet lined with parchment paper. Spoon the apple filling over half the pastry, leaving a border around the edge. Beat the eggs with 1 teaspoon water for the egg wash. Brush the edges of the pastry lightly with some of the egg wash. Fold the pastry over the filling to form a triangle. Press the edges to seal and enclose the filling completely, then press with the back of a fork to make a tight seal. Using a small knife, cut two 1/2-inch-long slits in the top of the crust. Brush the top with the egg wash.
Bake the turnover until the crust is puffed and golden and the juices are bubbling through the slits in the crust, about 25 minutes.
Sprinkle the turnover with the confectioners' sugar. Turn the oven to broil, and cook until the sugar is caramelized, watching it closely so it doesn't burn. This will give the turnover a browned crust. Carefully run a metal spatula under the turnover to loosen and transfer to a wire rack to cool slightly.
To serve, cut the turnover into wedges and serve warm with a dollop of cheese, yogurt or ice cream.