In a large bowl, combine 2 1/2 cups flour, salt, and 1 tablespoon sugar, then cut in the cold butter until the mixture resembles coarse crumbs.
Gradually add ice water, one tablespoon at a time, mixing until the dough just comes together, then divide into two disks, wrap in plastic, and refrigerate for at least 1 hour.
Preheat your oven to 425°F and roll out one disk of dough to fit a 9-inch pie plate, then transfer it to the plate.
In a large bowl, toss the sliced apples with 3/4 cup sugar, 2 tablespoons flour, cinnamon, and nutmeg until well coated.
Pour the apple mixture into the prepared pie crust and dot the top with small pieces of butter.
Roll out the second disk of dough and place it over the filling, trimming and crimping the edges to seal, then cut several slits in the top to vent.
Brush the top crust with the beaten egg wash and sprinkle with a little sugar if desired.
Bake for 15 minutes at 425°F, then reduce the temperature to 350°F and continue baking for 40-45 minutes until the crust is golden and the filling is bubbly.
Allow the pie to cool for at least 2 hours before slicing to let the filling set properly.