2 cups all-purpose flour
1 1/4 teaspoons salt, divided
3/4 cup plus, 2 tablespoons, sugar, divided
1 cup (2 sticks) unsalted butter, cold, cut in chunks
8 baking apples (such as Granny Smith or Golden Delicious), peeled, cored and sliced (about 8 cups)
2 tablespoons cider vinegar
1 teaspoon cinnamon
1/4 teaspoon nutmeg
2 tablespoons cornstarch
1 egg, separated
2 tablespoons rolled oats
1 tablespoon milk
In bowl of a food processor, pulse the flour, 1 teaspoon of the salt and 1 tablespoon of the sugar. Add the butter and pulse until mixture forms pea-sized chunks. Pulse in 1/3 cup water, 1 tablespoon at a time, just until mixture forms a dough. Wrap in plastic and refrigerate for 30 minutes.
Meanwhile, make filling. In a large deep skillet, combine the apples with the 3/4 cup sugar, the vinegar, cinnamon, nutmeg and remaining 1/4 teaspoon salt. Cook over medium heat, stirring and turning gently until apples are just tender, about 5 minutes. Stir together 2 tablespoons water and the cornstarch, then add to apples. Continue to cook until juices thicken, about 1 minute. Set aside to cool.
Heat oven to 350 F. Place a pizza stone, if available, in oven.
Divide dough into 2 pieces, 1 slightly larger than the other. On a floured surface, roll larger piece out to a 12-inch circle. Fold dough in half to make it easier to move, then transfer it to a 9-inch pie plate, unfolding it and allowing it to settle into bottom. Crust should overhang edge slightly.
In a small bowl, beat the egg white until slightly frothy. Brush it over surface of crust. Sprinkle the oats over bottom crust, then spoon apple filling into crust.
On a floured surface, roll smaller piece of dough into a 10-inch square. Using a pizza cutter or paring knife, cut dough into 1-inch strips. Weave strips in a lattice across top of apple filling. Alternatively, roll dough into a 10-inch circle and simply transfer to top of pie for a full crust.
Fold overhanging crust under bottom crust, gently tucking them into pie dish, then crimp edge. Beat the egg yolk in a small bowl with the milk and brush over surface of pie crust. Sprinkle with remaining 1 tablespoon sugar.
Bake in preheated oven on pizza stone or on a baking sheet set on rack in lowest position of oven until crust is golden-brown, about 45 minutes. Cool before cutting. Makes 8 servings.