Toss the apple chunks with the lemon juice, sugars, cinnamon, nutmeg, and salt.
Melt half the butter in a large non-stick pan over high heat. Add half of the apple mixture and fry, stirring occasionally, until the sugar mixture is a deep caramel color and apples are tender but not mushy (about 5-6 minutes). Remove to a bowl and repeat with the rest of the butter and apples. Stir in the vanilla. Set the apple mixture aside to cool completely.*
Roll the bottom crust out and place it in the pie dish. Put the egg white in a small bowl and give it a quick whisk. Brush the bottom crust with egg white and allow to dry (reserve the remaining egg white), then sprinkle with graham cracker crumbs.
Fill the pie with the cooled apple mixture, including any caramely sauce. Roll out the second crust and place it on top. Crimp as desired. Place the pie in the freezer for 15 minutes.
While the pie is chilling, place a sheet pan in the oven and preheat to 375 degrees.
Whisk a splash of milk into the remaining egg white. Take the pie out, and brush the top crust with the mixture, then sprinkle with turbinado sugar.
Place the pie on the preheated baking sheet on a lower rack in the oven. Bake for 45-55 minutes, or until the crust is golden brown. Remove from the oven and allow to cool for 30-45 minutes before slicing and serving.