1 1/2 sticks (3/4 cup) cold unsalted butter, cut into 1/2-inch cubes
1/4 cup cold vegetable shortening
1/2 teaspoon salt
4 to 6 tablespoons ice water
6 cups blackberries (1 3/4 lb)
1 to 1 1/4 cups granulated sugar
1/4 cup cornstarch
2 tablespoons unsalted butter, melted
2 tablespoons fresh lemon juice
2 tablespoons water
1 tablespoon quick-cooking tapioca
1 egg white, lightly beaten
1 tablespoon sanding (coarse) or granulated sugar
For pastry: blend together flour, butter, shortening, and salt in a bowl with your fingertips or a pastry blender (or pulse in a food processor) until most of mixture resembles coarse meal with rest in small (roughly pea-size) lumps. Drizzle 4 tablespoons ice water evenly over and gently stir with a fork (or pulse in processor) until incorporated. Gently squeeze a small handful: It should hold together without crumbling apart. If it doesn't, add more water, 1 tablespoon at a time, stirring (or pulsing) after each addition until incorporated, continuing to test. (Do not overwork dough or it will become tough.)
Turn dough out onto a work surface and divide into 2 portions. With heel of your hand, smear each portion once in a forward motion to help distribute fat. Gather each portion of dough and form it, rotating on a work surface, into a disk. Wrap disks separately in wax paper and chill until firm, at least 1 hour.
For filling: Preheat Oven to 400°F.