Preheat the oven to 400 degrees F (200 degrees C).
Peel and chop the apples into small pieces. I like to chop them up very well so that they cook quickly and evenly in the pan.
In a medium saucepan over medium heat, melt the butter. Once melted, add the chopped apples, maple syrup, and cinnamon. Stir until fully coated. Cook the apples, stirring often, until softened, about 10 minutes.
In a small bowl, stir together the cornstarch and 1 teaspoon of cold water until smooth. Pour this cornstarch mixture into the apples, stir, and cook until thickened. Remove from the heat to cool.
Unroll the pie crust on a floured surface and use a cookie cutter (about 2-2.5 inches in diameter) to cut out 12 circles. Set the scraps aside to use for the top layer of the mini pies.
Combine the scraps and re-roll the dough out. Use a pizza slicer or knife to cut out thin strips about 3 inches in length.
Place about 1 tablespoon of the apple mixture into the center of each circle. Place two crust strips horizontally and another two crust strips vertically over each mini pie. Press down on the sides and use a fork all around the edges of each pie to seal the crust in place.
Whisk together 1 egg and 1 tablespoon of water for the egg wash and brush this mixture on top of each pie.
Bake the pies for 15 minutes, or until golden brown.