In a food processor fitted with a metal blade, combine 1½ cups of flour, salt, and sugar. Pulse twice to incorporate.

Add the cold butter and shortening to the processor. Process for approximately 15 seconds, or until the mixture resembles cottage cheese curds with no visible dry flour.

Scrape the bowl with a rubber spatula and redistribute the dough evenly. Add the remaining 1 cup of flour and pulse 4 to 6 times until evenly mixed.

Transfer the mixture to a medium bowl. Sprinkle the cold vodka and cold water over the mixture. Using a rubber spatula, fold the dough until it begins to hold together and is slightly tacky.

Divide the dough into two equal portions and shape each into a 4-inch disk. Wrap each disk in plastic wrap and refrigerate for a minimum of 45 minutes and up to 2 days.

Remove one dough disk from the refrigerator. On a lightly floured surface, roll into a 12-inch circle, approximately ⅛-inch thick. Transfer to a 9-inch pie plate, allowing excess dough to hang over the edges.

Gently press the dough into the bottom and sides of the plate. Leave the overhang in place and refrigerate the pie shell until firm, approximately 30 minutes.

Adjust the oven rack to the lowest position. Place a rimmed baking sheet on the rack and preheat the oven to 400°F (200°C).

In a medium saucepan, combine 1 cup each of raspberries, blueberries, and blackberries. Mash the berries gently with a potato masher and cook over medium heat, stirring frequently, until the mixture thickens and reduces to about 1½ cups (approximately 8 minutes). Allow to cool slightly.

Place the grated apple in a clean kitchen towel and wring out excess moisture. Transfer to a large bowl.

Add the cooked berry mixture, remaining fresh berries, lemon juice, sugar, ground tapioca, and salt. Stir gently to combine.

Transfer the filling into the chilled pie shell and distribute evenly. Dot the surface with small pieces of butter.

Roll out the second disk of dough into a 12-inch circle. Using a pastry wheel, cut into twelve ¾-inch strips.

Arrange six strips horizontally across the top of the pie. Weave the remaining six strips vertically to form a lattice pattern.

Trim and fold the edges under themselves to seal. Flute using your fingers or press with a fork. Brush the top crust and edges with the prepared egg wash.

If the dough has softened, chill the assembled pie in the freezer for 10 minutes before baking.

Place the pie on the preheated baking sheet and bake for 30 minutes at 400°F.

Reduce the oven temperature to 350°F (175°C) and continue baking for an additional 30–40 minutes, or until the filling is bubbling and the crust is golden brown.

Transfer the pie to a wire rack and cool completely at room temperature for at least 4 hours before slicing and serving.