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).
While the dough rests, make the caramel sauce (For detailed instructions on making the sauce from scratch, see my caramel glaze post) or if you’re using store bought, warm a half cup of it in a sauce pan to loosen it enough to drizzle.
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.
Mix the brown sugar, dark brown sugar, flour, salt, nutmeg, cinnamon, and allspice together in a large bowl. Add the apples, melted butter, and vanilla and toss until combined.
Spread apples with their juices into a dough-lined pie plate, mounding them slightly in the middle
Drizzle with ½ cup of the salted caramel, reserving the rest for topping the pie
Loosely roll the remaining dough round around a rolling pin and gently unroll it onto filling. Trim the overhanging dough to ½ inch beyond the lip of the pie plate.
Pinch edges of the top and bottom crusts firmly together. Tuck the overhang under itself; the folded edge should be flush with the edge of the pie plate.
Crimp the dough evenly around the edge of the pie using your fingers. Cut four 2-inch slits on top of the dough. Brush surface with beaten egg white.
Adjust the temperature to 425°F, and bake until the crust is light golden brown, (about 20 minutes). Reduce oven temperature to 375°F, carefully cover with foil, using oven mitts rotate the baking sheet, and continue to bake until juices are bubbling and crust is deep golden brown (20-30 minutes more).
Let pie cool on wire rack to room temperature, about 4 hours. Serve drizzled with the remaining caramel sauce.
Notes
Avoiding A Soggy Bottom Pie Crust
My trick to making a crisp bottom is to adjust the oven rack to the lowest position, place a baking stone on the rack and then a rimmed baking sheet on the stone, and then heat the oven to 500°F.Once you’re ready to bake the pie, adjust the temperature to 425°F, and bake until the crust is light golden brown, (about 20 minutes). Reduce oven temperature to 375°F, carefully cover with foil, using oven mitts rotate the baking sheet, and continue to bake until juices are bubbling and crust is deep golden brown. If you don’t have a baking stone, a heated pan on the lowest position in the oven will help a whole lot too!