Prepare the Pie Crust: On a floured surface, roll out the chilled dough into a 12-inch circle. Place the dough into a 9-inch pie dish, easing it into the dish without stretching. Fold and tuck the edges, creating a ridge, and flute or crimp with a fork. Refrigerate until ready to use.
Make the Crumble Topping: In a food processor, pulse together the flour, brown sugar, cinnamon, and salt. Add the cold butter and pulse until the mixture forms large crumbs. Chill in the fridge while preparing the filling.
Cook the Berry Filling: Preheat the oven to 400°F. Place a rimmed baking sheet on the oven rack to catch any juices. In a large saucepan over medium heat, mash 3 ½ cups of the berries using a potato masher. Cook, stirring occasionally, until the mixture thickens and reduces to about 1 ½ - 1 ¾ cups. Let cool slightly.
Combine the Berry Filling: In a large bowl, mix the remaining 3 ½ cups of berries with both sugars, tapioca flour, salt, and lemon juice. Squeeze the grated apple in a clean kitchen towel to remove excess liquid, then add the apple to the uncooked berry mixture. Stir in the cooked berries until well combined.
Assemble the Pie: Pour the berry mixture into the chilled pie crust. Top with the chilled crumble mixture, spreading it evenly over the filling.
Bake: Place the pie on the preheated baking sheet and bake at 400°F for 30 minutes. Reduce the oven temperature to 350°F and continue baking for another 30-35 minutes, or until the juices are bubbling and the crumble is golden brown.
Cool: Transfer the pie to a wire rack and let cool to room temperature for 3-4 hours before serving.
Serve: Slice and serve at room temperature or slightly chilled. Enjoy!