In a large bowl, combine 1 cup blackberries, 1/4 cup sugar, the tapioca and lemon juice; mash with a potato masher. Stir in the strawberries, blueberries, remaining blackberries and 3/4 cup sugar. Let stand for at least 20 minutes, gently stirring once or twice.
Meanwhile, position a rack in the lower third of the oven and preheat to 400 degrees . In a large bowl, whisk together the flour and salt. Add 8 tbsp. butter and, with a pastry blender or 2 knives, cut the butter into the flour until crumbly. Continue working with your fingertips until the mixture is flaky. In a cup, combine the egg yolk with 1/4 cup ice water. Drizzle over the flour mixture and stir with a fork until incorporated and the dough is slightly moist but not wet. (Test by pinching some of the dough; if it doesn't stay together, sprinkle about 1 tsp. more ice water on the dry part and gather the dough into a mass.)
Transfer the dough to a lightly floured surface and, with a lightly floured rolling pin, roll out to a 13-inch round, rotating the dough occasionally to keep it even and to prevent it from sticking to the surface. Roll the dough up onto the pin and unroll over and into a 9-inch pie plate (see top right); center it and let the excess pastry drape over the edges. Gently press the dough into the pan (see middle right).
Spoon the berry mixture evenly into the pan, making sure to pour in all the juices that have collected in the bowl. Dot the top with the remaining 2 tbsp. butter. Gently fold the pastry up over the berries, pleating as you go around (see bottom right).
In a small bowl, beat the egg white with a fork until frothy. With a pastry brush, lightly glaze the pastry all over with the egg white. Sprinkle the dough with a few pinches sugar. Bake the pie until the berry filling is bubbling well, about 1 hour; check after 40 minutes and if the crust is browning too quickly, cover loosely with a sheet of foil. Let the pie cool completely, at least 4 hours, before cutting and serving with scoops of strawberry ice cream.