Preheat oven to 425 degrees.
Grate enough apples (about 2) to get about 1 cup of grated apple. Cut the rest of the apples into 4 sections (throw out the core) and slice the sections into 1/2 inch half moons.
In a large mixing bowl stir together the sliced apples, grated apple. and 1/4 cup flour. Set aside.
In a medium pan, over low heat, stir together the brown sugar, salt, butter, and water. Cook until hot, bubbly, and starting to thicken. Stir the sauce consistently so the sugar doesn't stick and burn on the bottom.
Once it bubbles remove from heat. Add milk and vanilla. Place back on the heat and cook until the sauce is bubble and thickened up. Pour over the apples and stir until all the apples are coated with sauce. Set aside.
Roll the dough out in-between the plastic sheets until it's the size you need. Press the dough into a pie dish.
Pour apple pie filling into the prepared pie shell. Place the second pie shell over the top into a lattice pattern or however you wish. Make sure the edges where the dough comes together is pinched and sealed.
Mix together the egg white and water and then brush the egg wash over the dough.
Bake the pie for 60 minutes. At 30 minutes place a piece of foil over top to prevent the crust from burning. Remove the foil with about 3-4 minutes left in cooking time.
Remove from the oven and let the pie sit for about 10 minutes before slicing. Serve warm, room temperature, or cold.