Prepare the pie crust: In a large bowl, mix the flour, sugar, and salt. Cut in the cold butter with a pastry cutter or fork until the mixture resembles coarse crumbs. Slowly add ice water and mix until the dough comes together. Divide into two discs, wrap in plastic wrap, and refrigerate for at least 30 minutes.

Preheat the oven to 425°F (220°C).

Prepare the filling: In a large bowl, toss the sliced apples with sugar, brown sugar, cinnamon, nutmeg, lemon juice, and flour until evenly coated.

Roll out one disc of dough on a lightly floured surface and fit it into the pie dish. Pour the apple filling into the crust and dot with butter.

Roll out the second disc of dough and place it over the apples. Trim the excess dough and crimp the edges to seal. Cut a few slits in the top crust to allow steam to escape.

Brush the top crust with the egg wash for a golden finish.

Bake for 45-55 minutes until the crust is golden brown and the filling is bubbly. If the edges brown too quickly, cover them with foil during baking.

Cool the pie on a wire rack for at least 2 hours before slicing and serving.