In a large bowl, whisk together the flour, sugar, and salt. Cut in the very 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. Be careful not to overwork the dough.

Divide the dough in half, flatten each half into a disc, wrap tightly in plastic wrap, and refrigerate for at least 30 minutes.

In a large bowl, combine the sliced apples, granulated sugar, brown sugar, flour, cinnamon, nutmeg, and lemon juice. Toss gently to coat the apples evenly.

Set the apple filling aside to macerate while you roll out the pie crust.

Preheat your oven to 400°F (200°C). On a lightly floured surface, roll out one disc of chilled dough into a 12-inch circle. Carefully transfer the dough to a 9-inch pie plate, trimming any excess and crimping the edges.

Pour the apple filling into the prepared pie crust. Dot the top of the filling with the small pieces of butter.

Roll out the second disc of dough and place it over the apple filling. Trim and seal the edges, then cut several slits in the top crust for ventilation.

Brush the top crust with the beaten egg and sprinkle with sugar.

Bake the pie for 15 minutes at 400°F (200°C), then reduce the oven temperature to 375°F (190°C) and bake for an additional 45-50 minutes, or until the crust is golden brown and the filling is bubbly.

If the crust starts to brown too quickly, cover the edges with aluminum foil. Let the pie cool completely on a wire rack before slicing and serving.