For The Pie Dough: Process 1 1/2 cups flour, salt, and sugar in food processor until combined, about two 1-second pulses. Add butter and shortening and process until homogenous dough just starts to collect in uneven clumps, about 15 seconds; dough will resemble cottage cheese curds and there should be no uncoated flour. Scrape bowl with rubber spatula and redistribute dough evenly around processor blade; Add remaining cup flour and pulse until mixture is evenly distributed around bowl and mass of dough has been broken up, 4 to 6 quick pulses. Empty mixture into medium bowl. Sprinkle vodka and water over mixture. With rubber spatula, use folding motion to mix, pressing down on dough until dough is slightly tacky and sticks together. Divide dough into 2 even balls and flatten each into 4-inch disk; wrap each in plastic wrap and refrigerate at least 45 minutes or up to 2 days. Remove 1 disk of dough from refrigerator and roll out on generously floured (up to 1/4 cup) work surface to 12-inch circle, about 1/8 inch thick. Roll dough loosely around rolling pin and unroll into pie plate, leaving at least 1-inch overhang on each side. Working around circumference, ease dough into plate by gently lifting edge of dough with one hand while pressing into plate bottom with other hand. Leave dough that overhangs plate in place; refrigerate while preparing filling until dough is firm, about 30 minutes. For The Filling: Adjust oven rack to lowest position, place rimmed baking sheet on oven rack, and heat oven to 400 degrees. Place 3 cups berries in medium saucepan and set over medium heat. Using potato masher, mash berries several times to release juices. (Note: If using frozen berries, see note in recipe description above). Continue to cook, stirring frequently and mashing occasionally, until about half of berries have broken down and mixture is thickened and reduced to 1 1/2 cups, about 8 minutes. Let cool slightly. Place grated apple in clean kitchen towel and wring dry; transfer apple to large bowl. Add cooked berries, remaining 3 cups uncooked berries, lemon zest, juice, sugar, tapioca, and salt; toss to combine. Transfer mixture to dough-lined pie plate and scatter butter pieces over filling. Roll out second disk of dough on generously floured (up to 1/4 cup) work surface to 11-inch circle, about 1/8 inch thick. Using 1 1/4-inch round biscuit cutter, cut round from center of dough; cut another 6 rounds from dough, 1 1/2 inches from edge of center hole and equally spaced around center hole. Roll dough loosely around rolling pin and unroll over pie, leaving at least 1/2-inch overhang on each side. Using kitchen shears, trim bottom layer of overhanging dough, leaving 1/2-inch overhang. Fold dough under itself so that edge of fold is flush with outer rim of pie plate. Flute edges using thumb and forefinger or press with tines of fork to seal. Brush top and edges of pie with egg mixture. If dough is very soft, chill in freezer for 10 minutes. Place pie on heated baking sheet and bake 30 minutes. Reduce oven temperature to 350 degrees and continue to bake until juices bubble and crust is deep golden brown, 30 to 40 minutes longer. Transfer pie to wire rack; cool to room temperature, at least 4 hours. Cut into wedges and serve.