In a food processor or blender, add in the flour, sugar, and cold diced butter. Pulse the flour and butter until the dough forms small pea-size pellets. Pour the flour into a large mixing bowl, then add in the water and begin to mix together. The dough should come together with added water, but should be wet, the dough should be flaky and crumbly. Once the dough comes together in a rough ball, place in saran or cling wrap and press into a thick disc. Place the dough in the fridge to chill for at least one hour.
Meanwhile, prepare the apple mixture by melting butter in a medium frying pan over medium heat. Add in the apples saute until the apples begin to soften, then add in the cinnamon and toss. To the apples, pour in the brown sugar and mix to combine, then add the cornstarch or arrowroot. The apple mixture should thicken, and once it does, remove from heat and set aside.
Take out the dough and place on a heavily floured surface. Roll out the dough until it forms a large rectangle, with the dough approximately ¼” thick. Using a pizza cutter, cut the dough into 10 even rectangles, then add 3-4 tbsp of apple mixture to five of the rectangles. Using the other pieces of dough, place on top of the pies and press down on all four sides using a fork to seal. Brush with the whisked egg and sprinkle with Demerara sugar. Place on a baking sheet lined with parchment paper. Bake at 400º for 22 minutes, let cool for 15 before serving.