Nutty rye flour and sharp cheddar cheese enrich buttery pie crust in this sweet-and-savory twist on classic apple pie. Tapioca starch thickens the filling, while keeping it clear and glossy.
Toss together apples and lemon juice in a large bowl. Stir together granulated sugar, light brown sugar, tapioca starch, nutmeg, cinnamon, and salt in a small bowl. Pour sugar mixture over apples, and toss with your hands until evenly distributed.
Stir together turbinado sugar and flour in a small bowl. Sprinkle bottom of unbaked pie crust with sugar-flour mixture. Spoon apple mixture into piecrust, being careful not to mound in center. Dot apples with butter cubes. Proceed with lattice and crimping instructions.
Preheat oven to 450°F. Remove pie from freezer, and place on a rimmed baking sheet lined with parchment paper. Brush lattice and crimped edge lightly with beaten egg. Bake until crust is evenly golden brown, 15 to 20 minutes. Without removing pie from oven, reduce oven temperature to 325°F, and continue baking until pie juices are bubbling in center, 50 minutes to 1 hour and 10 minutes. Remove baking sheet from oven, and transfer to a wire rack. Let pie cool completely before slicing, 4 to 6 hours.
Store leftover pie, covered, at room temperature up to 2 days.