1. Set the oven rack to the lowest position and place a baking sheet on the rack. Preheat the oven to 425 degrees Fahrenheit.
2. Select the larger disk of dough (it’s fine if they are the same size—in that case, use either one), sprinkle the top and bottom with flour, and roll it out on a generously floured surface to an 11-inch circle. Transfer the dough to a nine-inch pie plate and trim the overhang to one inch.
3. Mix the cranberry filling with a fork and spread it on the bottom of the pie crust. Spread the apple filling over the cranberry filling.
4. Sprinkle the top and bottom of the second disk with flour and roll it out on a generously floured surface to a 10-inch circle. Lay the dough over the filling, fold the overhang over, and crimp to seal.
5. You can use scraps of dough to make leaves, which you can attach with a bit of water as glue. Brush the beaten egg over the crust, sprinkle the cinnamon sugar over the top, and cut slits to vent.
6. Place the pie on the baking sheet and bake for 20 minutes. Reduce the heat to 350 degrees Fahrenheit and bake for another hour and a half, until the juices are bubbling. If the top crust browns too much, place a baking sheet on the shelf above the pie to slow the browning.