In a large bowl, whisk together the flour and salt. Cut in the cold 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 into a disk, wrap in plastic, and chill 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 ensure the apples are evenly coated.

On a lightly floured surface, roll out one disk of chilled dough into a 12-inch circle. Carefully transfer it to a 9-inch pie plate, trimming any excess dough to about a 1/2-inch overhang.

Pour the apple filling into the pie crust. Roll out the second disk of dough and place it over the filling. Trim the top crust to a 1/2-inch overhang, then fold both crusts under and crimp the edges to seal.

Cut several slits in the top crust for ventilation. Brush the top with beaten egg and sprinkle with turbinado sugar. Bake at 425°F (220°C) for 15 minutes, then reduce the temperature to 375°F (190°C) and bake for another 35-40 minutes, or until the crust is golden brown and the filling is bubbling.

Allow the pie to cool completely on a wire rack before slicing and serving. This ensures the filling sets properly.