In a small bowl, combine the sugars, flour and spices; set aside. In a large bowl, toss apples with lemon juice. Add sugar mixture; toss to coat.
Line a 9-in. pie plate with bottom crust; trim even with edge. Fill with apple mixture; dot with butter. Roll remaining crust to fit top of pie; place over filling. Trim, seal and flute edges. Cut slits in crust.
Beat egg white until foamy; brush over crust. Sprinkle with sugar. Cover edges loosely with foil.
Bake at 375° for 25 minutes. Remove foil and bake until crust is golden brown and filling is bubbly, 20-25 minutes longer. Cool on a wire rack.