Prepare the Pie Crust:
In a large bowl, mix the flour and salt.
Cut in the cold butter until the mixture resembles coarse crumbs.
Gradually add ice water, one tablespoon at a time, mixing until the dough comes together.
Divide the dough into two discs, wrap in plastic wrap, and refrigerate for at least 1 hour.
Prepare the Filling:
In a large bowl, combine the sliced apples, sugar, flour, cinnamon, nutmeg, lemon juice, and caramel sauce. Toss to coat the apples evenly.
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Assemble the Pie:
Preheat the oven to 400°F (200°C).
On a floured surface, roll out one disc of dough to fit a 9-inch (23cm) pie dish. Place the dough into the pie dish and trim the edges.
Pour the apple filling into the crust, spreading it evenly.
Roll out the second disc of dough and place it over the filling. Trim and crimp the edges to seal. Cut slits in the top crust to allow steam to escape.
In a small bowl, mix the beaten egg and milk. Brush the top crust with the egg wash.
Bake the Pie:
Place the pie on a baking sheet to catch any drips.
Bake for 20 minutes at 400°F (200°C), then reduce the temperature to 350°F (175°C) and bake for an additional 35–40 minutes, or until the crust is golden brown and the filling is bubbling.
Cool and Serve:
Allow the pie to cool on a wire rack for at least 2 hours before serving.
Drizzle additional caramel sauce over each slice when serving.