Preheat oven to 425°.
Lightly spoon 2 1/2 cups flour into dry measuring cups; level with a knife. Combine 2 1/2 cups flour and 1 teaspoon salt in a large bowl; cut in 1/2 cup butter with a pastry blender or 2 knives until mixture resembles coarse meal. Sprinkle surface with ice water, 1 tablespoon at a time; toss with a fork until moist and crumbly (do not form a ball).
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. Chill 30 minutes.
Roll 1 dough half, still covered, into a 12-inch circle. Roll remaining dough half, still covered, into an 11-inch circle. Chill 30 minutes.
Combine apple and juice in a bowl. Combine brown sugar, 1/4 cup granulated sugar, 3 tablespoons flour, 1 teaspoon cinnamon, and remaining 1/4 teaspoon salt in a small bowl. Sprinkle sugar mixture over apple mixture; 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, and dot with remaining 1 tablespoon butter; brush edges of dough lightly with water.
Remove top 2 sheets of plastic wrap from 11-inch dough circle; place, plastic wrap side up, over filling. Remove the remaining plastic wrap. Press edges of dough together. Fold edges under, and flute. Cut 4 (1-inch) slits in top of dough to allow steam to escape.
Combine remaining 1 tablespoon granulated sugar and remaining 1/8 teaspoon cinnamon in a small bowl. Brush top and edges of dough with egg white; sprinkle with sugar mixture. Place pie on a baking sheet. Bake at 425° for 25 minutes. Reduce oven temperature to 350° (do not remove pie from oven); bake an additional 30 minutes or until golden. Shield pie with foil, if necessary. Cool on a wire rack.