Preheat your oven to 375°F (190°C) and line a baking sheet with parchment paper.
In a large bowl, combine 2 cups of flour, 1 teaspoon of salt, and 1 teaspoon of sugar. Mix well.
Add the chilled, cubed butter to the flour mixture and use a pastry cutter or your fingers to mix until the mixture resembles coarse crumbs.
Gradually add ice water, one tablespoon at a time, mixing until the dough just starts to come together. Do not overwork the dough.
Divide the dough in half and shape each half into a disc. Wrap them in plastic wrap and refrigerate for at least 30 minutes.
While the dough chills, prepare the apple filling. In a bowl, mix the apple slices with lemon juice, brown sugar, cinnamon, nutmeg, and 2 tablespoons of flour. Set aside to let the flavors blend.
Roll out one disc of dough on a lightly floured surface into a rectangle, about ¼ inch thick. Transfer it to the prepared baking sheet.
Spread the apple filling evenly over the bottom crust.
Roll out the second disc of dough and place it over the apple filling. Trim any excess dough overhanging the edges and crimp the edges to seal.
Cut slits in the top crust to allow steam to escape. Brush the beaten egg over the top for a golden finish.
Bake in the preheated oven for 45 minutes, or until the crust is golden brown and the apples are tender.
Remove from the oven and allow to cool on a cooling rack for about 10 minutes before slicing.