Have ready and refrigerated one pastry-lined 9-inch pie pan and lattice strips to top (recipe below).
Whisk together 1 cup of the granulated sugar and the water in a medium saucepan, and cook over medium-low heat until the sugar is just dissolved. Add the butter and bring to a slow boil. Continue cooking over medium heat until the mixture turns a deep golden brown, almost copper. Remove from the heat and immediately but slowly add the heavy cream – be careful, the mixture will bubble rapidly and steam. Whisk the final mixture together well and set aside to cool while you prepare the apple filling.
Juice the lemons into a large mixing bowl, removing any seeds. Prepare the apples using an apple-peeling machine, or core, peel, and thinly slice them with a sharp knife or mandoline. Dredge the apple slices in the lemon juice. Sprinkle lightly with the remaining 2 tablespoons granulated sugar. Set aside to soften slightly and release some of the juices, 20 to 30 minutes.
In a small bowl, sprinkle the Angostura bitters over the raw sugar. Add the cinnamon, allspice, nutmeg, black pepper, kosher salt, and flour, and mix well. Add the prepared apples to the sugar-spice mixture, leaving behind any excess liquids. Gently turn the apples to evenly distribute the spice mix.
Tightly layer the apples in the prepared pie shell so that there are minimal gaps, mounding the apples slightly higher in the center. Pour a generous 1/2 cup to 3/4 cup of the caramel sauce evenly over the apples (use the larger quantity of sauce if you’d like a sweeter pie). Sprinkle with 1/4 teaspoon of the flake sea salt. Assemble the lattice on top of the pie and crimp the edges as desired.
Chill the pie in the refrigerator for 10 to 15 minutes to set the pastry. Meanwhile, position the oven racks in the bottom and center positions, place a rimmed baking sheet on the bottom rack, and preheat the oven to 400F.
Brush the pastry with the egg wash to coat, being careful not to drag the caramel onto the pastry (it will burn), and sprinkle with the desired amount of demerara sugar and flake sea salt. Place the pie on the rimmed baking sheet on the lowest rack of the oven. Bake for 20 to 25 minutes, or until the pastry is set and beginning to brown. Lower the oven temperature to 375F, move the pie to the center oven rack, and continue to bake until the pastry is a deep golden brown and the juices are bubbling, 30 to 35 minutes longer. Test the apples for doneness with a skewer or sharp knife; they should be tender and should offer just the slightest resistance.
Allow to cool completely on a wire rack, 2 to 3 hours. Serve slightly warm or at room temperature.
The pie will keep refrigerated for 3 days or at room temperature for 2 days.
Make the All-Butter Pie Crust:
Stir the flour, salt and sugar together in a large bowl. Add the butter pieces and coat with the flour mixture using a bench scraper or spatula. With a pastry blender, cut the butter into the flour mixture, working quickly until mostly pea-size pieces of butter remain (a few larger pieces are okay; be careful not to overblend).
Combine the water, cider vinegar, and ice in a large measuring cup or small bowl. Sprinkle 2 tablespoons of the ice water mixture over the flour mixture, and mix and cut it in with a bench scraper or spatula until it is fully incorporated. Add more of the ice water mixture, 1 to 2 tablespoons at a time, using the bench scraper or your hands (or both) to mix until the dough comes together in a ball, with some dry bits remaining. Squeeze and pinch with your fingertips to bring all the dough together, sprinkling dry bits with more small drops of the ice water mixture, if necessary, to combine. Shape the dough into a flat disc, wrap in plastic, and refrigerate for at least 1 hour, preferably overnight, to give the crust time to mellow. If making the double-crust version, divide the dough in half before shaping each portion into flat discs.
Wrapped tightly, the dough can be refrigerated for 3 days or frozen for 1 month.
Pro tip: If you are looking for the easiest way to roll out a pie crust, this pie board is the best $30 you will ever spend!