Preheat your oven to 425°F (220°C).
In a large bowl, combine flour and butter. Using a pastry cutter or your hands, mix until the mixture resembles coarse crumbs.
Gradually add ice water, mixing until the dough comes together.
Divide the dough in half, shape into discs, wrap in plastic wrap, and refrigerate for at least 30 minutes.
In another large bowl, combine apples, granulated sugar, brown sugar, cinnamon, nutmeg, lemon juice, and flour. Toss to coat the apples evenly.
Roll out one disc of dough on a floured surface to fit a 9-inch pie dish. Transfer the dough to the dish and trim excess.
Pour the apple mixture into the crust.
Roll out the second disc of dough and place it over the apples. Trim, seal, and crimp the edges. Cut slits in the top crust to allow steam to escape.
Brush the top crust with beaten egg.
Bake for 45-50 minutes, or until the crust is golden brown and the filling is bubbly.
Let cool before serving.
