Make the Pie Crust:

1. In a large mixing bowl, whisk together flour, sugar, and salt. Add the cold cubed butter and use a pastry cutter or two forks to cut the butter into the flour until the mixture resembles coarse crumbs with some pea-sized butter pieces remaining. This should take about 3-4 minutes.
2. Drizzle 6 tablespoons of ice water over the mixture and stir with a fork until the dough begins to come together. Add more water, one tablespoon at a time, if needed. The dough should hold together when pressed but not feel wet or sticky.
3. Divide the dough in half, shape each half into a flat disk, wrap tightly in plastic wrap, and refrigerate for at least 1 hour or up to 2 days. This resting time allows the gluten to relax and ensures a flaky crust.

Prepare the Apple Filling:

4. In a large bowl, combine sliced apples, both sugars, flour, cornstarch, cinnamon, nutmeg, and salt. Toss until apples are evenly coated. Add lemon juice and vanilla extract, then toss again. Let the mixture sit for 15-20 minutes while you roll out your dough – this allows the flavors to meld.

Assemble the Pie:

5. Preheat your oven to 425°F (220°C). Place a baking sheet on the lower rack to catch any drips.
6. On a lightly floured surface, roll out one disk of dough into a 12-inch circle, about ⅛-inch thick. Carefully transfer it to your 9-inch pie dish, gently pressing it into the bottom and sides. Trim any excess overhang to about ½ inch beyond the rim. Refrigerate while you prepare the top crust.
7. Roll out the second disk of dough into a 12-inch circle for the top crust. You can leave it whole for a traditional double-crust pie or cut it into strips for a lattice design.
8. Remove the pie dish from the refrigerator. Pour the apple filling into the bottom crust, mounding it slightly in the center. Distribute the small butter pieces over the top of the filling.
9. Place the top crust over the filling. If using a full crust, cut 4-5 slits in the top for steam to escape. Trim the edges, leaving a ½-inch overhang, then fold the edges under and crimp decoratively with your fingers or a fork to seal.
10. In a small bowl, whisk together the beaten egg and milk. Brush this egg wash generously over the entire top crust. Sprinkle with coarse sugar if desired for extra sparkle and crunch.

Bake the Pie:

11. Place the pie on the preheated baking sheet in the oven. Bake at 425°F for 20 minutes. Then reduce the temperature to 375°F (190°C) and continue baking for 35-45 minutes, until the crust is deep golden brown and the filling is bubbling through the vents.
12. If the edges begin to brown too quickly, cover them with foil or a pie crust shield after about 30 minutes of baking.
13. Remove the pie from the oven and let it cool on a wire rack for at least 4 hours before slicing. This cooling time is essential – it allows the filling to set properly so you get clean slices instead of a runny mess.