Peel and thinly slice the apples, then toss them with the lemon juice to prevent them from browning. Each apple should be cut into approximately 32 slices.
Add the cranberries, sugar, cornstarch, and cinnamon to the bowl of apple slices and toss until well coated.
Place half of the pastry on a lightly floured work surface and roll it into 11 or 12-inch diameter circle. Transfer the pastry to a large parchment-lined cookie sheet.
Arrange half of the apple slices and cranberries on the circle of pastry, leaving a border of about 3 inches all the way around.
Fold the pastry over the cranberry apple filling, then brush it with the beaten egg and sprinkle the pastry and filling with demerara sugar.
Bake the cranberry apple galette at 400 F for 35 minutes, or until the pastry is a rich golden brown. Let it cool on the tray for 5 minutes, then let it finish cooling on a wire rack. Cover the filling with foil if it's starting to brown too much.
Repeat steps 3-6 with the other half of the pastry and filling.
Serve the cranberry apple galette warm with vanilla ice cream or crème anglaise.