Preheat the oven to 425°F with a rack in the lower third. Place a baking sheet on the rack below to catch drips.
Peel, core and slice the apples about ¼ inch thick. Toss with both sugars, the flour, cinnamon, nutmeg, lemon juice and salt.
Fit one crust into a 9-inch pie plate. Pile in the apple filling, mounding it slightly in the center, and dot with the butter.
Lay the second crust over the top. Trim the excess, fold the edges under and crimp to seal. Cut 4–5 small slits in the top to let steam escape.
Brush the top with the beaten egg and sprinkle with sugar.
Bake for 20 minutes. Reduce the oven to 375°F and bake 35–45 minutes more, until the crust is deep golden and the juices bubble through the vents. Cover the edges with foil if they brown too quickly.
Cool for at least 3 hours before slicing so the filling can set.