In a large bowl, whisk together the flour and salt. Cut in the cold, cubed butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining. Gradually add ice water, one tablespoon at a time, mixing until the dough just comes together. Divide the dough in half, flatten each half into a disk, wrap in plastic, and chill in the refrigerator for at least 30 minutes.

In a large bowl, combine the peeled, cored, and sliced apples with granulated sugar, brown sugar, flour, cinnamon, nutmeg, and lemon juice. Toss gently until the apples are evenly coated.

Preheat your oven to 425°F (220°C). On a lightly floured surface, roll out one disk of chilled pie dough into a 12-inch circle and carefully transfer it to a 9-inch pie plate. Pour the apple filling into the crust and dot the top with the small pieces of butter. Roll out the second dough disk and place it over the apple filling. Trim and crimp the edges of both crusts together to seal, then cut a few slits in the top crust to allow steam to escape. For an optional golden finish, whisk the egg and water to create an egg wash, and brush it lightly over the top crust. Bake for 15 minutes, then reduce the oven temperature to 375°F (190°C) and continue to bake for 35-45 minutes more, or until the crust is golden brown and the filling is bubbly. Let the pie cool completely on a wire rack before serving.