Preheat oven to 450°.
To prepare crust, lightly spoon 2 cups flour into dry measuring cups; level with a knife. Combine 1/2 cup flour, ice water, and vinegar, stirring with a whisk until well blended to form a slurry. Combine remaining 1 1/2 cups flour, powdered sugar, and 1/2 teaspoon salt in a large bowl; cut in shortening with a pastry blender or 2 knives until mixture resembles coarse meal. Add slurry; toss with a fork until flour mixture is moist.
Divide dough in half. Gently press each half into a 4-inch circle on 2 sheets of overlapping heavy-duty plastic wrap; cover with 2 additional sheets of overlapping plastic wrap. Roll 1 dough half, still covered, into a 12-inch circle. Roll other dough half, still covered, into an 11-inch circle. Chill dough 10 minutes or until plastic wrap can be easily removed.
To prepare filling, combine the apples and lemon juice in a large bowl. Combine 2/3 cup sugar, 3 tablespoons flour, cinnamon, nutmeg, and 1/8 teaspoon salt in a small bowl. Sprinkle sugar mixture over apples; toss well to coat.
Remove top 2 sheets of plastic wrap from 12-inch dough circle; fit dough, plastic wrap side up, into a 9-inch deep-dish pie plate coated with cooking spray, allowing dough to extend over edge. Remove remaining plastic wrap. Spoon filling into dough; brush edges of dough lightly with water.
Remove top 2 sheets of plastic wrap from 11-inch dough circle; place, plastic wrap side up, overfilling. Remove remaining plastic wrap. Press edges of dough together. Fold edges under, and flute. Cut 4 (1-inch) slits into top of pastry using a sharp knife. Brush top and edges of pie with egg white; sprinkle with 1 tablespoon sugar.
Place pie on a baking sheet; bake at 450° for 15 minutes. Reduce oven temperature to 350° (do not remove pie from oven), and bake an additional 40 minutes or until golden. Cool on a wire rack.