Prepare the crust: Roll out the dough into a 12-inch round on a lightly floured surface. Ease into a 9-inch pie plate, fold the overhanging dough under itself and crimp the edge with your fingers. Chill 30 minutes.
Position a rack in the lower third of the oven and preheat to 350 degrees F. Line the dough with foil and fill with pie weights or dried beans. Bake until golden around the edge, about 20 minutes. Remove the foil and weights and continue baking until golden all over, 10 to 15 more minutes. Transfer to a rack and let cool completely.
Meanwhile, make the filling: Peel the apples; cut into 1/2-inch pieces. Toss with the granulated sugar and lemon juice in a bowl. Melt the butter in a large skillet over medium-high heat. Add the apples and cook, stirring occasionally, until slightly softened, 10 to 12 minutes. Add the flour, cinnamon and salt and stir until the juices thicken, 2 minutes. Remove from the heat and fold in the berries; let cool completely. (The filling can be made up to 2 days ahead; cover and refrigerate.)
While the filling cools, make the topping: Whisk the flour, granulated sugar, baking powder, baking soda, nutmeg and salt in a large bowl. Cut in 3 tablespoons butter with your fingers until it is in pea-size pieces. Add the buttermilk; stir with a wooden spoon to make a shaggy dough.
Spoon the filling into the crust; dot with the remaining 1 tablespoon butter. Drop large spoonfuls of the topping over the filling and sprinkle generously with turbinado sugar. Place the pie on a rimmed baking sheet. Transfer to the oven and bake until the topping is golden, 50 minutes to 1 hour. (Cover the pie with foil if it is browning too quickly.) Transfer to a rack and let cool until set, about 3 hours.