For the pie crust: Combine the flour, sugar and salt in the bowl of the food processor and give it a quick pulse to blend the ingredients. Add the butter and pulse until the mixture looks like crumbly sand. Add the cheese and pulse again until combined and cheese is worked into the mixture and cut into small chunks. Turn the mixture out into a bowl, drizzle in 6 tablespoons cold water, and mix in with a rubber spatula until the flour is moistened. Add another tablespoon of water, only if needed. Divide the dough in half, shape each half into a ball and flatten to a disc. Rest the dough to relax the gluten, 30 minutes.
For the filling: Combine the Granny Smith apples, Braeburn apples, sugar, flour, light brown sugar, cinnamon and nutmeg in a large bowl and toss to combine.
Preheat the oven to 425 degrees F. Adjust the baking racks to the center of the oven.
Roll out 1 disc of the dough on a lightly floured surface to a 12-inch round. Transfer to a 9-inch pie dish. Add the apples in an even layer and dot with the butter. Lightly brush the edges with egg wash. Roll out the next dough round, slightly smaller than the first. Cover the top of the apples with the second pie dough, pressing the edges to seal the crust. Trim the crust so you are left with a 1-inch overhang. Tuck the crust underneath itself so it's flush with the edge of the pie plate. Crimp the edges decoratively. Brush the pie with the egg wash and sprinkle with the turbinado sugar. Make 5 air slits in the center of the pie. Place the pie on baking sheet and bake for 25 minutes.
Lower the oven temperature to 350 degrees F and bake for 45 minutes longer. Cool to room temperature, at least 3 hours, before slicing.