In a large bowl, combine flour and salt. Add the chilled butter and mix until the mixture resembles coarse crumbs. Stir in ice water, a tablespoon at a time, until the mixture forms a ball. Divide the dough in half, shape into disks, wrap in plastic wrap, and refrigerate for at least 4 hours or overnight.

Prepare the Filling

In a large bowl, toss the sliced apples with granulated sugar, brown sugar, cinnamon, nutmeg, and lemon juice. Set aside for about 30 minutes to allow the juices to mix.

Assemble the Pie

Preheat the oven to 425°F (220°C). Roll out one dough disk on a floured surface to fit a 9-inch pie pan. Place the dough into the pan. Fill with the apple mixture, then roll out the second disk and cover the filling. Seal the edges and cut slits for steam to escape. Brush the top with beaten egg.

Bake the Pie

Bake in the preheated oven for 15 minutes. Reduce the temperature to 350°F (175°C) and continue baking for an additional 35 to 45 minutes, until the apples are soft and the crust is golden brown. Allow to cool before serving.