In a large bowl, whisk together the flour, sugar, and salt.
Add the cold, cubed butter to the flour mixture. Using a pastry cutter or your fingers, cut the butter into the flour until the mixture resembles coarse crumbs, with pea-sized pieces of butter.
Gradually add ice water, one tablespoon at a time, mixing gently until the dough starts to come together. You may need 6 to 8 tablespoons of ice water. Be careful not to overwork the dough.
Divide the dough into two equal parts, shape them into discs, wrap in plastic wrap, and refrigerate for at least 1 hour (or up to 2 days).
Prepare the apple filling:
Peel, core, and slice the apples into ¼-inch thick slices.
In a large bowl, toss the apple slices with lemon juice, sugars, cinnamon, nutmeg, flour (or cornstarch), and vanilla extract. Let the mixture sit for 10 minutes to allow the flavors to combine.
Roll out the pie crust:
Preheat your oven to 400°F (200°C).
On a lightly floured surface, roll out one of the chilled dough discs to fit a 9-inch pie dish (about 12 inches in diameter). Carefully transfer the rolled-out dough into the pie dish, allowing the excess to hang over the edges.
Roll out the second dough disc to make the top crust.
Assemble the pie:
Pour the apple filling into the prepared pie crust, making sure to arrange the apples in an even layer.
Dot the top of the apple filling with small pieces of butter.
Place the second rolled-out dough over the apples. Trim any excess dough, leaving about a 1-inch overhang.
Fold the overhanging dough under itself to seal the edges, then crimp the edges using your fingers or a fork for decoration.
Cut a few small slits in the top crust to allow steam to escape while baking.
Egg wash and baking:
Brush the top of the pie with the beaten egg for a shiny, golden crust. Sprinkle with coarse sugar if desired.
Place the pie on a baking sheet to catch any drips, and bake in the preheated oven for 45-55 minutes, or until the crust is golden brown and the apple filling is bubbling.
If the edges start to brown too quickly, cover them with aluminum foil or a pie shield halfway through baking.
Cool and serve:
Remove the pie from the oven and allow it to cool for at least 2 hours before slicing, to allow the filling to set.
Tips:
Use a mix of tart and sweet apples for a more complex flavor.
You can make the dough ahead of time and freeze it for later use.
Serve your apple pie with vanilla ice cream or whipped cream for an extra treat.
