Preheat the oven to 425 degrees F (220 degrees C).

Melt butter in a small saucepan over medium-low heat; stir in flour to make a smooth paste. Whisk white sugar, brown sugar, water, and lemon juice into butter mixture. Bring to a simmer, whisking constantly, until sauce thickens, about 3 minutes. Stir 6 blackberry halves into sauce, reduce heat to low; continue cooking while you complete remaining steps, stirring sauce occasionally.

Line a 9-inch pie dish with 1 pastry crust. Arrange apple slices and remaining blackberries in crust. Cut remaining pastry crust into 1-inch wide strips; weave into a lattice top over filling. Pinch strip ends to crust edges to seal. Spoon blackberry sauce over pie; let it soak through crust. Transfer pie to a baking sheet.

Bake in the preheated oven for 15 minutes. Reduce oven temperature to 350 degrees F (175 degrees C); continue baking until crust is golden, and sauce bubbles, 35 to 45 more minutes.