In a medium saucepan, combine the apples, sugar, brown sugar, cinnamon, nutmeg, and lemon juice.
Stir in the ¼ cup of water, stirring as it heats. Allow the mixture to become hot, then add in the cornstarch and water.
Continue stirring the apple mixture for about 5-7 minutes. The apple mixture will begin to thicken, and once it does, remove the pan from heat and set it aside to let it cool.
Prepare the oven, and preheat to 400 degrees.
Meanwhile, in a small bowl, stir together the brown sugar and cinnamon for the topping of the pies.
Set that bowl aside until the pies are prepared for the oven.
Roll out each pastry puff sheet into a square, about 12x12. Cut each square into 8 even rectangle pieces, for a total of 16 pieces from both sheets.
Place four of the rectangle pieces on a baking sheet, lined with parchment paper.
Spoon some of the apple pie filling onto the center of each piece, leaving about ¼ inch space on the edges.
Gently brush the edges with some of the egg wash, then cover the apples with another rectangle piece.
Pull the dough so that it reaches each corner and the rectangles line up. Press the pieces together with the end of a fork. Repeat with all four pieces.
Lightly brush the top of each pie with the egg wash.
Slice four small slits across the top of each piece, and then sprinkle a spoonful of the cinnamon and brown sugar mixture on top of each pie.
Place in the oven and bake for about 16-18 minutes, until the pastry begins to turn golden brown.
Repeat and assemble the other four pies, or place on a second baking sheet, lined with parchment paper, and bake at the same time.
This recipe makes 8 copycat apple pies.