Roll out 1 layer of pie dough and place it in a 9-inch glass or ceramic pie pan. Flute the edges, and prick the bottom of the dough with a fork. Whisk the egg white and water together in a small bowl, then brush it all over the bottom pie crust. Set aside.
In a large saucepan set over medium heat, melt the butter. Stir in the flour and cornstarch to form a paste. Add water, sugars, spices, and salt and bring to a boil, stirring often. Reduce temperature to low and simmer until a caramel-like sauce has formed. Cool slightly.
Place prepared apples in a large bowl. Pour the caramel sauce over the apples and toss to coat.
Transfer the apples to the pie crust, mounding them slightly. Cover with a lattice crust. Brush the remaining egg wash on the lattice work. Use a pie shield or aluminum foil to gently cover the edges of the crust, then place the pie pan on a large baking sheet before placing it in the oven.
Bake 55 minutes, then carefully brush the top lattice work with milk or cream. Continue baking for about 10 more minutes until the top is nicely browned. Serve with ice cream and caramel sauce, if desired.
Notes
If the crust is browning too quickly, cover loosely with strips of foil.