Preheat your oven to 375°F (190°C). Position your rack in the lower third of the oven.
Roll out your pie crust to approximately ⅛-inch thickness on a lightly floured surface. Transfer it to a 9-inch pie dish, gently pressing it into the bottom and sides. Trim any excess dough, leaving about a ½-inch overhang, then fold and crimp the edges. Refrigerate the prepared crust for 15 minutes.
In a large bowl, gently toss the sliced apples, blackberries, granulated sugar, cornstarch, lemon juice, cinnamon, nutmeg, salt, and vanilla extract. Fold rather than stir vigorously to maintain the integrity of the blackberries. Allow this mixture to rest for 10 minutes.
While the filling rests, combine the flour, oats, brown sugar, cinnamon, and salt in a medium bowl. Add the cold butter cubes and use a pastry cutter or your fingertips to work the butter into the dry ingredients until the mixture resembles coarse crumbs with some pea-sized pieces of butter remaining. For extra crunch, chill the completed crumble in the refrigerator for 10 minutes.
Remove the chilled pie crust from the refrigerator and pour in the fruit filling, mounding it slightly in the center. Evenly sprinkle the crumble topping over the filling, covering it completely and pressing very lightly to adhere.
Place the pie on a baking sheet lined with parchment paper and bake for 20 minutes at 375°F. After 20 minutes, reduce the oven temperature to 350°F (175°C) and continue baking for an additional 25-30 minutes, or until the filling is bubbling around the edges and the crumble topping is golden brown. If the edges begin to brown too quickly, cover those areas loosely with aluminum foil.
Allow the pie to cool completely on a wire rack for at least 2-3 hours before slicing. This cooling period is crucial as it allows the filling to set properly.