Step 1

Make the Crust: Process 1½ cups of the flour, the sugar and salt together in a food processor until combined, about 2 one-second pulses. Add the butter and shortening and process until a homogenous dough just starts to collect in uneven clumps, about 7 to 10 seconds (the dough will resemble cottage cheese curds with some very small pieces of butter remaining, but there should be no uncoated flour). Scrape down the sides and bottom of the bowl with a rubber spatula and redistribute the dough evenly around the bowl. Add the remaining 1 cup flour and pulse until the mixture is evenly distributed around the bowl and the mass of dough has been broken up, 4 to 6 quick pulses. Empty the mixture into a medium bowl.

Step 2

Sprinkle the vodka and water over the mixture. With a rubber spatula, use a folding motion to mix, pressing down on the dough until it is slightly tacky and sticks together. Squeeze dough gently until cohesive and divide into two equal balls. Flatten each into a 4-inch disk, wrap in plastic wrap and refrigerate at least 45 minutes, or up to 2 days.

Step 3

Remove one disk of dough from the refrigerator and roll out on a generously floured (up to ¼ cup) work surface to a 12-inch circle. Roll the dough loosely around a rolling pin and unroll into a 9-inch pie plate, leaving at least a 1-inch overhang on each side. Working around the circumference, ease the dough into the plate by gently lifting edge of the dough with one hand while pressing into the plate bottom with other hand. Leave any dough that overhangs plate in place. Wrap dough-lined pie plate loosely in plastic wrap and refrigerate until dough is firm, about 30 minutes. Roll other disk of dough into 12-inch circle on lightly floured counter, then transfer to parchment paper–lined baking sheet; cover with plastic and refrigerate for 30 minutes.

Step 4

Prepare and Bake the Pie: Adjust oven rack to lowest position, place rimmed baking sheet on rack, and preheat oven to 500 degrees F.

Step 5

In a large bowl, whisk together the sugar, tapioca, salt, nutmeg, cinnamon, and allspice. Add the apples and lemon juice and toss until combined. Spread apples with their juices into dough-lined pie plate, mounding them slightly in middle.

Step 6

Loosely roll remaining dough round around rolling pin and gently unroll it onto filling. Trim overhang to ½-inch beyond lip of pie plate. Pinch edges of top and bottom crusts firmly together. Tuck overhang under itself; folded edge should be flush with edge of pie plate. Crimp dough evenly around edge of pie using your fingers. Cut four 2-inch slits in top of dough. Brush surface with beaten egg white and sprinkle evenly with remaining 1 tablespoon sugar.

Step 7

Place pie on heated baking sheet, lower oven temperature to 425 degrees F, and bake until crust is light golden brown, about 25 minutes. Reduce oven temperature to 375 degrees, rotate baking sheet, and continue to bake until juices are bubbling and crust is deep golden brown, 30 to 35 minutes longer. Let pie cool on wire rack to room temperature before serving, about 4 hours.