To prepare crust: Whisk whole-wheat flour, all-purpose flour, 2 tablespoons sugar and salt in a large bowl. Cut butter into small pieces and, with your fingers, quickly rub them into the dry ingredients until the pieces are smaller but still visible. Add sour cream and oil; toss with a fork to combine with the dry ingredients. Sprinkle water over the mixture. Toss with a fork until evenly moist. Knead the dough with your hands in the bowl a few times¿the mixture will still be a little crumbly. Turn out onto a clean surface and knead a few more times, until the dough just holds together. Divide the dough in half and shape into 5-inch-wide disks. Wrap each in plastic and refrigerate for 1 hour.
To prepare filling: Combine apples, brown sugar, lemon juice, 1 teaspoon cinnamon, nutmeg, allspice and salt in a large bowl. Reserving 4 cups, transfer the rest of the apple mixture to a Dutch oven. Cook over medium heat, stirring, until the apples are tender and beginning to break down, about 10 minutes. Remove from the heat, stir in the reserved apples and 2 tablespoons flour; let cool for about 30 minutes.
To assemble & bake pie: Position a rack in the lower third of the oven; preheat to 425°F.
Remove the dough from the refrigerator; let stand for 5 minutes to warm slightly. Roll one portion between sheets of parchment or wax paper into a 13-inch circle. Peel off the top sheet and invert the dough into a 9 1/2-inch deep-dish pie pan. Peel off the remaining paper. Scrape the filling into the crust. Roll the remaining portion of dough between sheets of parchment or wax paper into another 13-inch circle. Peel off the top sheet of paper and invert the dough onto the fruit. Peel off the remaining paper. Trim the crust so it overhangs evenly. Tuck the top crust under the bottom crust, sealing the two together and making a plump edge. Flute the edge with your fingers. Combine remaining 1 teaspoon of granulated sugar and the remaining 1/4 teaspoon cinnamon in a small bowl. Brush the crust with egg white and sprinkle with the cinnamon-sugar. Cut 6 steam vents in the top crust.
Bake the pie for 20 minutes; reduce the oven temperature to 375° and continue baking until the crust is golden brown and the filling is bubbling, 25 to 35 minutes more. Let cool on a wire rack for about 1 1/2 hours before serving.
The best apple pie filling combines ¿saucy¿ apples (that cook down to a saucelike texture) and ¿shapely¿ apples (that hold their shape after baking). Pick one variety from each category¿here we used McIntosh and Granny Smith. Saucy: McIntosh, Cortland, Macoun, Mutsu (Crispin), Paula Red or Empire. Shapely: Granny Smith, Golden Delicious, Jonathon, Jonagold, Northern Spy or Ida Red.
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