In a large bowl, whisk together the gluten-free flour, salt, and 1 tablespoon of sugar.
Incorporate the cold cubed butter using a pastry cutter or fork until the mixture resembles coarse crumbs with some pea-sized pieces.
Add ice water 1 tablespoon at a time, mixing until the dough starts to come together into a ball; divide dough into two disks, wrap in plastic, and refrigerate for 30 minutes.
Preheat your oven to 425°F (220°C) and lightly grease a 9-inch pie dish.
In a separate large bowl, toss the sliced apples with 3/4 cup granulated sugar, brown sugar, cornstarch, cinnamon, nutmeg, and lemon juice until evenly coated.
Roll out one disk of chilled dough between two sheets of parchment paper to a 12-inch circle and carefully transfer it to the pie dish.
Fill the crust with the apple mixture and dot the top with the remaining 1 tablespoon of butter pieces.
Roll out the second disk of dough, place it over the apples, and crimp the edges of the top and bottom crusts together to seal.
Cut several small slits in the top crust to allow steam to escape, then brush the entire top with the beaten egg wash.
Bake at 425°F for 15 minutes, then reduce the oven temperature to 375°F (190°C) and continue baking for 40 to 45 minutes until the crust is golden brown and filling is bubbling.
Remove from the oven and allow the pie to cool completely on a wire rack for at least 3 hours to ensure the filling sets before slicing.