Preheat oven to 400°.
To prepare crust, lightly spoon flour into dry measuring cups; level with a knife. Combine 1 tablespoon flour and cheese in a small bowl; toss well. Place mixture in freezer for 10 minutes.
Combine 1/4 cup flour and 1/2 cup ice water in a small bowl, stirring with a whisk until well blended to form a slurry. Combine 1 3/4 cups flour and 1/2 teaspoon salt in a large bowl; cut in butter with a pastry blender or 2 knives until mixture resembles coarse meal. Stir in cheese mixture. Add slurry; toss with a fork. Sprinkle surface with up to 1 tablespoon ice water, tossing with a fork until moist and crumbly (do not form a ball). Mixture will seem slightly dry.
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 each dough half, still covered, into a 12-inch circle. Chill 30 minutes or until plastic wrap can be easily removed.
To prepare filling, combine apple and lemon juice in a large bowl. Combine 3 tablespoons flour, brown sugar, and next 5 ingredients (brown sugar through dash of salt) in a small bowl. Sprinkle brown sugar mixture over apples; toss well to coat.
To assemble pie, remove top sheets of plastic wrap from 1 dough circle; fit dough, plastic wrap side up, into a 10-inch deep-dish pie plate coated with cooking spray, letting dough extend over edge of plate. Remove remaining plastic wrap. Spoon filling into crust. Brush edges of crust lightly with water. Remove top sheets of plastic wrap from remaining dough circle; place, plastic wrap side up, on apple mixture. Remove remaining plastic wrap. Press edges of dough together; fold edges under, and flute. Cut 3 (1-inch) slits in top of pastry using a sharp knife.
Combine egg white and 1 tablespoon water; brush over top and edges of pie. Place pie on a baking sheet; bake at 400° for 45 minutes or until golden. Cool on a wire rack.