Make the pie crust: In a large bowl, whisk together 2.5 cups flour and 1 tsp salt. Cut in the cold, cubed butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs with some pea-sized butter pieces. Gradually add ice water, 1 tablespoon at a time, mixing until the dough just comes together. Do not overmix. Divide dough in half, flatten into discs, wrap in plastic wrap, and refrigerate for at least 30 minutes.

Prepare the filling: In a large bowl, combine the sliced apples, granulated sugar, brown sugar, cinnamon, nutmeg, and 2 tablespoons of flour. Toss gently to coat the apples evenly. Stir in lemon juice.

Assemble the pie: Preheat oven to 425°F (220°C). On a lightly floured surface, roll out one disc of dough into a 12-inch circle. Carefully transfer it to a 9-inch pie plate. Trim edges, leaving about a 1-inch overhang.

Add filling: Pour the apple mixture into the pie crust. Dot the top with small pieces of butter.

Top crust: Roll out the second disc of dough. Place it over the filling. Trim the edges to match the bottom crust overhang. Crimp the edges together to seal, using a fork or your fingers. Cut several slits in the top crust to allow steam to escape.

Egg wash and bake: In a small bowl, whisk the egg and milk for the egg wash. Brush the top of the pie with the egg wash and sprinkle with a little extra sugar, if desired.

Bake for 15 minutes at 425°F (220°C). Then reduce the oven temperature to 375°F (190°C) and continue baking for 35-45 minutes, or until the crust is golden brown and the filling is bubbling. If the crust starts to brown too quickly, cover the edges with foil.

Cool and serve: Let the pie cool on a wire rack for at least 2-3 hours before slicing and serving. This allows the filling to set properly. Serve warm, perhaps with a scoop of vanilla ice cream.