Make the crust: Combine the flour, granulated sugar and salt in a 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 the cheese is in small chunks and combined into the mixture. Turn the mixture out into a bowl, drizzle in 6 tablespoons cold water and mix 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 into a disk. Wrap in plastic and refrigerate 30 minutes.
Meanwhile, make the filling: Combine the apples, flour, granulated sugar, brown sugar, cinnamon and nutmeg in a large bowl and toss to combine.
Put a baking sheet on the middle oven rack and preheat to 425 degrees F. Roll out 1 disk of dough into a 12-inch round on a lightly floured surface. Transfer to a 9-inch pie dish. Add the apples in an even layer and dot with the butter. Lightly brush the edges of the crust with the egg wash. Roll out the other disk of dough slightly smaller than the first. Place on top of the pie and press the edges together to seal. Trim the crust so you are left with a 1-inch overhang. Tuck the crust edge under itself and crimp. Brush the top crust with the egg wash and sprinkle with the turbinado sugar. Make several air slits in the center of the pie.
Place the pie on the hot baking sheet and bake for 25 minutes. Reduce the oven temperature to 350 degrees F and bake until golden brown and bubbling, 1 hour to 1 hour 15 minutes longer. Let cool at least 3 hours before slicing.