In a large bowl, combine flour, salt, and sugar for the crust. Cut in cold butter until the mixture resembles coarse crumbs.

Gradually add ice water, a tablespoon at a time, mixing until the dough just comes together. Divide in half, shape into disks, wrap, and refrigerate for at least 30 minutes.

In another large bowl, combine sliced apples with both sugars, cinnamon, nutmeg, allspice, salt, flour, and lemon juice. Toss until evenly coated.

Stir in the caramel sauce and gently mix to combine.

Preheat oven to 400°F (200°C). Roll out one dough disk and fit into a 9-inch pie dish.

Pour half of the apple filling into the crust and drizzle with half the caramel sauce. Repeat with the remaining apples and caramel. Dot the top with butter pieces.

Roll out the second dough disk and place over the pie. Trim and crimp the edges, and cut slits on top for steam to escape.

Brush the top crust with beaten egg and sprinkle with coarse sugar if desired.

Bake for 45-50 minutes, or until the crust is golden and the filling is bubbly. Cover edges with foil if browning too quickly.

Cool for at least 1 hour before slicing. Drizzle with extra caramel sauce before serving.