Preheat oven to 375.
Peel, core, and slice apples into thin, uniform slices, about 1/4 inch thick. As you are slicing the apples, toss them in a bowl with apple juice (enough to cover the apple slices) to prevent browning.
Once the apples are all sliced, drain the apples in a colander placed on top of a bowl. Save the apple juice.
In a large sauce pan, whisk the dry ingredients together - the sugars, cornstarch, cinnamon, salt, and nutmeg.
Add 1 1/2 cups apple juice (use the apple juice used to cover the apple slices) to the dry ingredients. The mixture will be thick, but as the sugars melt, it will thin out.
Bring to a boil, stirring with a whisk until it reaches the consistency of a thick glaze. Whisk in a little more apple juice if it becomes too thick.
Stir drained apple slices into the glaze.
Cover and reduce heat to medium-low. Cook until the apples just start to soften and become limp. Stir every couple of minutes. This should take about 5-7 minutes.
Pour hot apple filling into an uncooked pie crust. Do not overfill the pie crust. The filling should be level with the top of the pie tin.
Lightly moisten the outside rim of the pie crust by dipping your finger in water and "painting" around the edge of the crust. This will help to seal in the juices so it doesn't make a mess in your oven.**
Place the top crust over the pie. Trim off any excess of the top crust that comes over the edge of the pie tin with a butter knife.
Seal the top crust to the bottom crust by crimping the edge as shown.
Slit vents in the top of the crust in a decorative pattern.
In a small bowl, whisk egg white until it is foamy. Using a pastry brush, brush the top of the pie with the egg white.
Bake for 50 minutes. After 20 minutes check the pie. If the crimped edge of the crust is becoming too brown, use strips of aluminum foil to cover the rim of the pie for the remaining cooking time.