Preheat oven to 375 degrees. In a small mixing bowl, mix 1/2 white sugar, brown sugar, flour, cinnamon, ginger, and nutmeg. Set aside.
In a large mixing bowl, toss apples with lemon juice. Sprinkle in sugar mixture and toss to evenly coat.
Lay one pie crust in bottom of 9-inch pie pan, trimming the pastry even with the edge of the pan. Spoon in the apple mixture into the pie and place butter piece on top.
Roll the remaining crust out. Set crust over the filling and trim as necessary to fit the top of the pie. Seal, flute the edges, and cut slits in the pastry.
Beat egg white until foamy; brush over pastry. Sprinkle with sugar. Cover edges loosely with foil.
Bake for 25 minutes. Remove foil and bake until crust is golden brown and pie filling is bubbling, about 20 minutes more. Cool on wire rack.