If necessary, roll one of the dough portions to about ⅛" thick and at least 1" wider than the pie plate on a lightly floured surface. Loosely roll it around a rolling pin and gently unroll it onto a 9-inch pie plate, letting excess dough hang over the edge.
Ease dough into the plate by gently lifting the edge of dough with 1 hand while pressing into the plate bottom with the other. Leave any dough that overhangs, and wrap the dough-lined pie plate loosely in plastic wrap and refrigerate until dough is firm, about 30 minutes (resting is only necessary if the dough has been rolled, but it should be wrapped and placed in the fridge until needed).
Roll other dough portion to also be about ⅛" thick and at least 1" wider than the pie plate on a lightly floured surface then transfer it to a parchment paper-lined baking sheet; cover with plastic and refrigerate for 30 minutes (again, resting is only necessary if the dough has been rolled, but it should be wrapped and placed in the fridge until needed).
Adjust oven rack to lowest position, place if you have one, place a baking stone on the rack and a rimmed baking sheet on the stone (if you don't have a stone, just use the baking sheet), and heat oven to 500°F.
Sprinkle the old-fashioned oats evenly over the bottom of the dough-lined pie plate. This step helps absorb excess moisture from the apples.
In a large bowl, mix together the brown sugar, dark brown sugar, flour, salt, nutmeg, cinnamon, and allspice. Add the sliced apples, melted butter, and vanilla. Toss until the apples are thoroughly coated with the mixture.
Spread the apple mixture evenly over the oats in the pie crust, mounding slightly in the center to create a domed shape.
Roll out the second piece of pie dough and place it over the apples. Trim the edges, seal them with the bottom crust, and flute as desired. Cut slits into the top crust to allow steam to escape.
Brush the top crust with a beaten egg white to create a golden and glossy finish.
Place the pie on the baking stone, reduce the oven temperature to 425°F (220°C), and bake for 25 minutes. Then, lower the temperature to 375°F (190°C) and bake for an additional 35–40 minutes, or until the crust is golden brown and the filling is bubbling.
Allow the pie to cool completely on a wire rack for at least 2 hours before slicing. This helps the filling set properly.
Notes
If you have juicy apples, place half of the apples in a sauté pan and heat over medium heat, until some of the juices cook out and the apples soften (about 10 minutes). Then remove the apples from the pan, use a slotted spoon and stir them into the remaining uncooked apples to cool.