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To make the salted caramel:
Cook the sugar and water together over low heat until just dissolved. Add the butter and bring to a slow boil. Continue cooking at a low boil until the mixture turns a deep, golden brown color, almost copper.
Cook's Note: This process can take awhile depending on the heat source. Keep an eye on it, if the caramel begins to smoke, you've burned it and you'll have to start over.
Once the mixture has turned a copper color, remove it from the heat and immediately add the heavy cream - the mixture will bubble rapidly and steam - be cautious as the sugar will be very hot.
Whisk the final mixture together well over low heat and sprinkle in the sea salt. Set the caramel aside while you prepare the apple filling.
To make the apple filling seasoning:
In a large measuring cup or small mixing bowl, combine the sugar, flour, cinnamon, allspice, nutmeg, and Angostura bitters. Sprinkle this mixture over the apples in the mixing bowl. Use your hands to gently mix and coat the apple slices.
To assemble the pie:
Preheat the oven to 375 to 400 degrees F (depending on the hotness of your oven).
Gather your rolled pie crust, salted caramel, and apple mixture. Begin by layering 1/3 of the apples in the bottom of the crust so that there are minimal gaps. Pour 1/3 of the caramel over the apples. Add 1/3 of the apples and caramel for a second layer, and then add a third layer of apples, and then the caramel again. Cook's Note: Save a small portion of the caramel to pour on top once the lattice is assembled.
Assemble the lattice crust and flute the edges of the crust. Pour the last bit of caramel on top. Brush the crust with the beaten egg and lightly sprinkle with raw sugar and sea salt.
Bake the pie on a baking sheet larger than the pie pan for 20 minutes (otherwise the caramel will bubble over and burn on the bottom of your oven). Reduce the oven temperature to 325 to 350 and bake for 25 to 35 minutes. You can test the apples for doneness with a long toothpick or small knife. The apples should be just soft.
Let the pie cool, then slice and enjoy.