Preheat the oven to 375°F with a baking tray.
In a large bowl, combine the sugar, lemon juice, and apple slices. Let stand for 15 minutes. In a second bowl, combine the flour, nutmeg, and cinnamon.
Line a 9-inch pie plate with one piece of pie dough, allowing it to droop over the edges slightly. Drain the sliced apples in a colander, then put them back in the bowl. Add the flour and spice mixture, and toss to combine. Place the apples on top of the dough evenly, one layer at a time.
Place the second pie dough over the apple mixture and gently press down so it sits well on top of the apples. Use a knife to make four slits or cut a hole the size of a quarter in the center of the dough so the steam can escape. Trim and crimp the edges, then brush the top of the pie with the beaten egg, making sure to get all the sides.
Bake the pie on the baking tray for 35 minutes. Rotate, reduce the heat to 325°F, and bake an additional 25 minutes until golden brown. Remove from the oven and place on a cooling rack for one hour before slicing.