Prepare the pie crust dough (or use store-bought) and chill it for at least 1 hour before rolling out.
Preheat the oven to 400°F (204°C) and position the rack in the center.
In a large bowl, toss sliced apples with sugars, flour, cinnamon, nutmeg, lemon juice, and zest until evenly coated.
Roll out one pie crust to 12 inches in diameter and ⅛-inch thick, then lay it into a deep dish pie plate.
Add the apple filling to the crust, discard excess juices, and cover with the second rolled-out pie crust.
Trim and seal the edges, fold them under neatly, and cut slits in the top crust for steam to escape.
Brush the pie with egg wash, sprinkle with sugar, and cover the edges with foil or a pie shield.
Bake at 400°F for 25 minutes, then reduce to 375°F, remove the shield, and bake for 30-35 minutes until golden and bubbly. Cool for 3 hours before serving.