Pie Dough:

In a large bowl, mix flour, sugar, and salt.
Add cold butter and shortening; use a pastry cutter to blend until mixture forms pea-sized lumps.
Gradually mix in ice water until the dough is cohesive but not wet; adjust with more flour or water if needed.
Divide the dough into two halves, wrap each in plastic, and refrigerate for 2 hours or overnight.
Apple Filling:

Peel and slice apples into 1/8 inch thick pieces. Toss in a bowl with lemon juice and sugar to prevent browning.
Let apples macerate in the refrigerator for about 2.5 hours, or until they release 1/4 cup of juice.
In a saucepan, melt butter, then whisk in flour, apple juice, sugars, cinnamon, nutmeg, and vanilla. Cook to a caramel-like consistency.
Toss apples in the sauce, then layer into the chilled pie crust, creating a mound.
Top with a lattice crust, seal edges, and freeze the pie while preheating the oven to 425°F.
Brush with egg wash, sprinkle with raw sugar, and bake. Start at 425°F for 15 minutes, then reduce to 350°F and bake for another 50-60 minutes. Cover with foil if browning too quickly.