Have ready and refrigerated one pastry-lined 9-inch pie pan and lattice strips to top. This pie shell may be purchased in dairy section of your market.

Caramel sauce
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.

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 on a 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.

Baking
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 400˚F. 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 375˚F, 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.

Note: The pie will keep refrigerated for 3 days or at room temperature for 2 days.