For the dough: In the bowl of a food processor, combine the flour, 2 tablespoons sugar, 1 teaspoon salt, and the butter. Pulse until the butter forms pea-size crumbles. Pour in 1⁄4 cup ice-cold water and pulse briefly until dough begins to clump. Scrape the dough onto a work surface, knead until it just comes together, and mold into a ball. Halve the dough and shape each half into a 1⁄2-inch-thick disk. Wrap each disk separately in plastic wrap and refrigerate for 1 hour.
For the filling: In a large bowl, toss the blackberries and blueberries with the remaining 1⁄2 cup sugar, 1⁄2 teaspoon salt and the tapioca pearls until evenly combined. Let stand for 20 minutes.
Heat the oven to 400°. On a lightly floured work surface, roll one dough disk into a 14-inch circle and fit it into a 10-inch glass pie dish. Stir the berries to redistribute the juices and the sugar, then scrape into the pie dish. Roll the second dough disk into a 12-inch circle and place over the berries. Trim, seal, and crimp the edges of the dough and cut four slits in the top. Brush the dough all over with the beaten egg.
Place the pie on a parchment paper—lined baking sheet and bake until the crust begins to turn light golden brown, 15 minutes. Reduce the oven temperature to 325° and continue baking until the berries are bubbling and the crust is golden brown, 45 minutes. Let the pie cool before serving.