Blackberry Pie With Lattice Crust

Name: Blackberry Pie With Lattice Crust
Ingredients: 2½ cups all-purpose flour; ¾ cup (6 oz.) cold unsalted butter, cut into ½ cubes; ¼ cup (2 oz.) cold vegetable shortening; ½ tsp. Diamond Crystal or ¼ tsp Morton kosher salt; 4 Tbsp. ice water, plus more; 6 cups blackberries (1¾ lb.); 1–1¼ cups sugar; ¼ cup cornstarch; 2 Tbsp. unsalted butter, melted; 2 Tbsp. fresh lemon juice; 2 Tbsp. water; 1 Tbsp. quick-cooking tapioca, ground into a powder; 1 egg white, lightly beaten; 1 Tbsp. coarse sanding sugar or granulated sugar; Whipped cream, vanilla ice cream, or cinnamon ice cream (for serving)