To prepare the crust:
Combine the flour, sugar, and salt in the bowl of a food processor. Pulse in the butter until pea-sized clumps exist throughout. Pulse in the cheese to combine barely. Pour the vinegar into your ice water and stir. Add about 5 tablespoons of the water mixture into the bowl of the processor and pulse a bit. Add water a tablespoon more at a time until the mix begins to come together into a shaggy dough. Don’t overwork it. Dump the dough out onto a clean surface and bring it together into a ball. Cut the dough in half and flatten each piece into a disk. Wrap in plastic wrap and allow to chill in the fridge for at least 2 hours.
To prepare the pie:
Preheat the oven to 425 degrees. Use a floured rolling pin to roll out half of the dough onto a floured surface into a circle about an inch wider than your pie plate on all sides. Roll the dough back onto the pin and then unroll it into the pie dish. Brush a thin layer of the egg white wash onto the pie dough and place in the freezer while you prep your filling. Combine the sliced apples with the sugar, brown sugar, flour, lemon juice, spice, vanilla, and salt. Toss to evenly coat the apples. Gently ease the dough into the edges of the pie dish and dump the filling into the prepared pie dish. Roll out the second half of dough onto a flour surface and use it to cut lattice strips or top your pie. Cut a few vent strips into the top of the pie if you opt out of a lattice and crimp the edges of the pie together as desired. Brush a thin layer of the egg white wash on top of the pie dough and place in the oven. Decrease the oven heat to 400 degrees and bake for 20 minutes. Decrease the heat to 375 and bake for an additional 30 minutes or until the filling is bubbling under the crust. You can top the pie crust with a sheet of aluminum foil if it begins to look too dark. Allow to cool completely prior to cutting!