In a mixing bowl, combine flour and butter until mixture resembles coarse crumbs. Gradually add ice water, mixing until dough forms. Divide dough in half, wrap in plastic wrap, and refrigerate for at least 30 minutes.
Roll out one half of the dough on a floured surface to fit a baking sheet. Transfer to the baking sheet.
In a large bowl, combine apples, sugar, cinnamon, nutmeg, and flour. Spread the apple mixture over the dough in the baking sheet.
Roll out the second half of the dough and place it over the apple mixture. Seal the edges and cut slits in the top crust. Brush with beaten egg.
Bake for 45 minutes or until the crust is golden brown. Let cool before serving.