Make the pie crust recipe and set it in the fridge to chill. Meanwhile, make the caramel sauce and set aside to cool a bit.
Preheat oven to 400 degrees F.
Roll out the bottom crust on a lightly floured surface. Transfer it to a 9-inch pie plate and set in the fridge to stay cold while working on the filling.
To make the filling, peel and slice the apples into ¼-inch slices. Toss them in a large bowl with the flour, sugar, lemon juice, lemon zest, cinnamon, cloves, and nutmeg until evenly coated.
Transfer the apple pie filling to the bottom crust, mounding the apples slightly in the center. They will bake down as they cook. Drizzle ½ cup of the caramel sauce over the top of the apples.
Roll out the top crust too cover the pie. You can use a solid crust with slits cut into it to vent, or cut the top crust into strips and weave into a lattice crust. There is also a streusel recipe in the notes below if you prefer a Dutch caramel apple pie.
Beat the egg with a tablespoon of water, then brush the egg wash over the top crust. Sprinkle with coarse sugar, if desired.
Bake for 20 minutes, then decrease the oven temperature to 375 degrees F and continue to bake for another 40-50 minutes, or until the crust is golden brown and the filling is bubbling inside. If the top crust starts getting too dark around the edges, cover them with aluminum foil or a silicon pie protector if you have one.
Let the pie cool for at least 4 hours before serving so that the filling has time to set up. To serve, drizzle the whole pie or each slice of pie with additional caramel sauce.