To make the crust: Combine flours, sugar, and salt in a large bowl. Cut in butter and shortening with 2 knives or a pastry blender until mixture resembles coarse crumbs. Mix in sour cream and lemon juice until dough comes together; some lumps are okay. Divide dough in half and shape into 2 disks. Wrap each in plastic wrap and chill in the refrigerator for 1 hour.

Preheat the oven to 350 degrees F (175 degrees C). Grease a 9-inch pie dish.

Roll 1 dough disk out to fit the prepared 9-inch pie dish. Roll out remaining dough disk for top crust; set aside.

To make the filling: Place apples into crust in the pie dish. Stir together sugar, flour, and nutmeg in a small bowl; sprinkle over apples. Drizzle with lemon juice.

Brush outer edge of bottom crust with beaten egg. Cover with top crust; trim and crimp edges together. Brush top with beaten egg and cut 4 steam vents.

Bake in the preheated oven until golden brown, about 45 minutes. Allow pie to completely cool before serving.