Filling Adjust oven rack to lowest position, place a rimmed baking sheet on oven rack, and preheat oven to 400°F. Place 3 cups berries in a medium saucepan set over medium heat. Using a potato masher, mash berries several times to release juices. Continue to cook, stirring frequently and mashing occasionally, until about half of berries have broken down, mixture is thickened and reduced to 1½ cups, about 8 minutes. Cool slightly.

Place grated apple in a clean kitchen towel and wring dry. Transfer the apple to a large bowl. Add the cooked berries, remaining 3 cups uncooked berries, lemon zest, lemon juice, sugar, tapioca, and salt; toss to combine. Transfer the mixture to dough-lined pie plate and scatter butter pieces over filling.

Roll out second disk of dough on a generously floured work surface to an 11-inch circle. Cut out rounds of dough, shapes, or prepare a lattice crust. Roll dough loosely around rolling pin and unroll over pie, leaving at least ½-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 the edges or press with tines of fork to seal. Whisk together the egg and water, and brush the top and edges of pie with egg mixture. Place in the freezer for at least 10 minutes.

Place the pie on the heated baking sheet and bake for 30 minutes. Reduce oven temperature to 350°F and continue to bake until the juices bubble and crust is deep golden brown, 30-40 minutes longer. Transfer pie to wire rack; cool to room temperature, at least 4 hours. Cut into wedges and serve.