How to Make It
Preheat the oven to 400° and position racks in the bottom and center thirds of the oven.
Whisk together 1 cup of granulated sugar and 1/4 cup of 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.
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 large 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 if you like a sweeter pie). Sprinkle with 1/4 teaspoon of the flaky 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.
Brush the pastry with the egg wash to coast, being careful not to drag the caramel in to the pastry (it will burn). Sprinkle with the desired amount of Demerara sugar and flaky sea salt.
Place the pie on a rimmed baking sheet and transfer to 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. Transfer the pie to a wire rack and let cool for 3 hours. Serve warm or at room temperature.