Mix apples, 1/4 cup brown sugar, white sugar, 1/4 cup flour, and cinnamon together in a bowl.
Combine 3/4 cup flour, oats, and 2/3 cup plus 2 tablespoons brown sugar in a bowl. Drizzle melted butter over flour mixture; stir until crumbly.
Pile apple mixture into the pie shell. Scatter crumbly topping over apples.
Bake in the preheated oven until apples start to soften, about 15 minutes. Reduce oven temperature to 350 degrees F (175 degrees C). Continue baking until apples are soft and topping is browned, about 40 minutes more.