We love using a variety of apples in our pie; it adds both flavor and texture and makes every bite a little different. Vodka in the pie crust makes the dough easier to work with, and since the alcohol burns off during baking, it doesn't impart any flavor. But feel free to use bourbon or apple brandy instead to complement the filling.
For the dough: Pulse the flour, granulated sugar and salt in a food processor to combine. Add the butter and process until the largest pieces of butter are pea-size. Transfer to a large bowl.
Stir the vodka, vinegar and 1/4 cup ice water in a small bowl (or 1/2 cup plus 1 tablespoon ice water if you're not using vodka). Drizzle the wet ingredients over the dough, and then mix with a fork until shaggy pieces form. Knead the dough in the bowl with your hands a couple of times until it comes together (it will look quite dry, which is fine). Transfer the large clumps of dough to a work surface. Drizzle 1 tablespoon ice water over any remaining smaller bits of dough in the bowl and knead again to bring it together. If the remaining dough is still too dry to come together, add more ice water in 1 tablespoon increments. Add to the dough on the work surface and press together into a single mass, incorporating any dry bits. Then pat the dough into a 1-inch-thick block. Divide the block into 4 pieces with a bench scraper or knife. Stack the pieces on top of one another, tucking any unincorporated dry bits in between the layers. Flatten the dough into a 3/4-inch-thick block. Repeat this process (cutting, stacking and flattening) three more times; this creates layers of butter in the dough that produces a wonderful flaky, almost puff pastry-like crust.
Divide the dough in half and form into 1-inch-thick discs; wrap each tightly in plastic wrap. Refrigerate at least 3 hours or preferably overnight. The dough can be made 3 days ahead. Keep it refrigerated or freeze it for up to 3 months.
To fill and assemble: Peel, core and slice the apples into 1/2-inch wedges. Toss the apples, brown sugar, flour, butter, lemon juice, cinnamon, salt, allspice and nutmeg in a large bowl. Let the fruit sit for 30 minutes to extract the juices.
Meanwhile, soften one dough disc at room temperature for 5 minutes. Line a rimmed baking sheet with parchment paper. Lightly flour your work surface. Roll the dough out to 1/8-inch thick (about 16 inches in diameter). Transfer to the prepared baking sheet and refrigerate. Soften the second disc at room temperature for 5 minutes. Then roll it out to the same thickness and diameter as the first disc.
Carefully transfer the second disc to a 9-inch pie dish. Lift the edges so the dough slumps down into the dish. Press the dough firmly against the sides and bottom of the dish. Trim the edges, leaving about a 1-inch overhang. Refrigerate for 5 minutes to firm up. Remove the first disc from the refrigerator and let it soften for 5 minutes.
Arrange a rack in the center of the oven and preheat to 425 degrees F. Line another rimmed baking sheet with foil and place it on the center rack.
Scrape the apple filling into the pie dish, creating a mound in the center. Beat the egg with 1 teaspoon water in a small bowl and brush the edges of the dough. Place the other disc over the filling. Trim the edges, leaving about a 1/2-inch overhang. Fold the bottom edge up and over the top edge; press together to seal. Crimp the edge and brush the top with egg wash. Sprinkle with the demerara sugar. Cut several vents in the top evenly spaced. Freeze the pie for 10 minutes.
Put the pie dish on the preheated baking sheet. Bake for 5 minutes, and then reduce the heat to 375 degrees F. Continue to bake for 45 minutes, and then loosely tent with foil. Continue baking until the crust is a deep golden brown and the juices are thick and vigorously bubbling, 35 to 45 minutes longer. The juices will start to bubble at around 75 minutes, but they will thicken and bubble faster in the last 15 minutes; don't be tempted to pull it out until the bubbles are really going. (If using a clear pie dish, check underneath to make sure the bottom crust is evenly browned.) Transfer the pie to a wire rack and let it cool at least 4 hours before serving. (Yes, it smells amazing, and yes, people love warm pie. But if you don't give it time to set up properly, the filling will be runny when you cut into it.)
When measuring flour, we spoon it into a dry measuring cup and level off excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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