Grated apple is the secret to a syrupy pie filling--it releases more pectin (a natural thickener) than the sliced apples alone. A quick salted caramel intensifies the apple flavor without making the dessert too sweet.
To prepare crust, weigh or lightly spoon flours into dry measuring cups; level with a knife. Place 1 1/2 cups all-purpose flour, pastry flour, 1 tablespoon sugar, and 1/4 teaspoon salt in a food processor; pulse 3 to 4 times to combine. Add 6 tablespoons butter and canola oil; pulse 8 to 10 times or until mixture resembles coarse meal. Add 1/4 cup ice water; pulse until dough begins to form a ball. Turn dough out onto a work surface; divide and shape into 2 (8-inch) disks. Wrap in plastic wrap, and refrigerate 1 hour or overnight.
To prepare filling, combine brown sugar, 2 tablespoons butter, 1 tablespoon water, and 3/8 teaspoon salt in a small saucepan over medium-high. Cook 7 minutes or until mixture begins to bubble and thicken, stirring frequently. Remove pan from heat; stir in milk and vanilla. Return pan to medium-high; bring to a boil. Cook 1 minute, stirring constantly. Remove pan from heat; cool slightly.
Preheat oven to 425°F.
Unwrap 1 dough disk and place between 2 large sheets of parchment paper. Roll to an 11-inch circle. Place into a 9-inch pie plate coated with cooking spray. Refrigerate until ready to use.
Grate enough apple quarters to equal 1 cup. Combine grated apple and 1 tablespoon all-purpose flour in a bowl. Cut remaining apple quarters into 1/2-inch-thick slices; combine with remaining 3 tablespoons flour. Add grated apple mixture to sliced apple mixture. Pour cooked brown sugar mixture over top; stir to coat. Spoon filling into prepared pie shell.
Unwrap remaining dough disk; roll to an 11-inch circle. Cut circle into 10 (1/2-inch) strips. Arrange in a lattice pattern over apple mixture, sealing dough strips to edge of crust. Combine remaining 1 tablespoon water and egg white in a bowl; brush lattice and dough edges. Bake at 425°F for 45 minutes. Cover loosely with foil, and bake at 425°F for 15 more minutes or until filling is bubbly and crust is golden brown. Cool at least 20 minutes before serving.