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

(Melt butter in a 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 is thick, about 3 minutes. Stir 6 blackberry halves into sauce, reduce heat to low, and continue cooking while you complete remaining steps. Stir sauce occasionally.)

(Line a 9-inch pie dish with 1 pastry crust. Place apple slices and remaining blackberries into crust. Cut remaining pastry crust into strips, about 1-inch wide. Use strips to weave a lattice top over apple-blackberry filling; pinch ends of strips to edge of bottom crust to seal. Spoon hot blackberry sauce over pie, letting it soak through the openings in top crust. Place pie onto a baking sheet.)

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