In a large bowl, mix flour, salt, and sugar. Add the cold, cubed butter.

Using a pastry cutter or your fingers, mix until the mixture resembles coarse crumbs.

Add ice water, one tablespoon at a time, mixing until the dough comes together.

Divide the dough into two balls, flatten into disks, wrap in plastic wrap, and refrigerate for at least 1 hour.

Peel, core, and slice the apples into thin wedges.

In a large bowl, combine the apple slices, sugar, flour, cinnamon, nutmeg (if using), lemon juice, and vanilla extract. Toss until apples are well coated.

Preheat your oven to 425°F (220°C).

On a floured surface, roll out one disk of dough into a circle about 12 inches in diameter. Place it in a 9-inch pie pan.

Roll out the second disk and cut it into strips (about ¾ inch wide) for the lattice top.

Pour the apple filling into the bottom crust, spreading it evenly.

Lay the strips of dough over the filling in a lattice pattern: place one strip across the center, then alternate strips over and under it to create the lattice. Trim the excess dough and fold the edges of the bottom crust over the lattice strips, crimping to seal.

In a small bowl, mix the beaten egg with milk. Brush the egg wash over the lattice and sprinkle with sugar.

Bake in the preheated oven for 20 minutes. Reduce the temperature to 350°F (175°C) and bake for an additional 30-40 minutes, or until the apples are tender and the crust is golden brown.

Allow the pie to cool for at least 2 hours before slicing. This helps the filling set.