In a large bowl, combine 1 cup blackberries, 1/4 cup sugar, the tapioca, lemon juice, and lemon zest and mash with a potato masher. Stir in strawberries, blueberries, raspberries, remaining blackberries, and 3/4 sugar. Let stand for at least 20 minutes, gently stirring once or twice.
Meanwhile, preheat oven to 400*F. Roll pie crust to 13" if you make your own. If you're using pre-rolled crusts, I would recommend putting the nine inch crust in and then attaching the additional crust. Leave the excess crust hanging over the edge of a 9" pie pan.
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 butter. Gently fold the pastry up over the berries, pleating as you go around.
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 generously with sugar.
Bake the pie at 400*F until the berry filling is bubbling, about an 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 four hours, before cutting and serving with scoops of ice cream (or frozen yogurt.