Prepare pie crust doughs through step 4, forming into a disc, wrapping, and chilling for at least 45 minutes before proceeding (or you may use store bought pie crusts). While they chill, preheat oven to 425F (220C) and prepare your filling.
In a large mixing bowl, whisk together sugars, flour, cinnamon, and salt.
Add apples, lemon juice, and vanilla extract and toss together until apples are thoroughly coated with the mixture.
Remove one pie dough from the fridge and transfer to a lightly floured surface. Roll into a 12” (30cm) circle and then arrange into a 9” (23cm) or 9 ½” pie plate.
In a separate, small bowl whisk together egg and 1 teaspoon of water until thoroughly combined to create your egg wash. Remove pie plate from the fridge and lightly brush the pie dough with egg wash.
Briefly stir apples and pour into the prepared pie plate. Place butter pieces evenly over the apples. Set aside.
Remove your second pie crust dough from the refrigerator and roll it into a 12” (30cm) circle. Cover the apple pie and with the second crust. Fold the excess crust underneath and flute or crimp the edges. Cut vents into the top of the pie dough (I do a large “X” then some smaller dashes).
Using your prepared egg wash, brush the top pie crust with an even layer and sprinkle all over with coarse sugar.
Place on a rimmed, foil-lined baking sheet and transfer to 425F (220C) preheated oven. Bake for 15 minutes, then reduce the oven temperature to 350F (175C) and continue to bake for another 45-55 minutes, until apples are tender and crust is golden (use a wooden skewer to test that the apples are tender enough to be easily pierced). Keep an eye on the crust and if the edges begin to over-brown, lightly tent them with foil (I just take a large piece of foil, cut a hole in the center, then rest it over the pie so the edges are covered).
Allow to cool for at least several hours before cutting into and enjoying. Apple pie is best served warm with a generous scoop of vanilla ice cream!