Roll both crusts to fit a 9 inch pan, about 1/8 inch thick. Chill the crust while the rest of the pie is being prepared.
To begin the caramel, combine water in sugar in a medium sauce pan over medium heat. Bring to a boil and cook until mixture turns amber.
Stir in butter, being careful as the butter will cause it to bubble and the mixture is hot.
Once melted, slowly drizzle in the heavy cream while stirring.
Once cream is completely incorporated, allow to cook for 30 seconds to 1 minute more. The mixture will rise as it cooks. If it gets too close to the top, remove it from heat.
Stir in salt, then set aside to cool.
Peel, core, and slice apples into thin wedges. Toss apple slices in lemon juice and bitters.
In a mixing bowl, combine sugar, flour, lemon zest, cinnamon, allspice, nutmeg, and salt.
Add sugar and spices to apples, toss to coat.
Preheat oven to 400 F with the rack in the center. It may also be a good idea to line the lower rack with a foil or put in a baking sheet lined with foil to catch pie drippings.
Fit the bottom pie crust into the pan.
Pour about 1/2 the apple mixture into the pie crust. Pour on about 1/3 the caramel onto the apples. Then add the remaining apples, topping with another 1/3 of the caramel. Set the last 1/3 of the caramel aside to be served with the pie slices.
Place the top pie crust over the pie, trimming excess, and pressing the top and bottom crusts together. Cut small vents in the top crust.
Beat the egg and brush over the pie crust. Sprinkle on sugar and salt, if desired.
Bake for 20 minutes, then reduce the heat to 375 F and bake until golden brown, about another 40 minutes. If the top of the pie is browning too quickly, cover it loosely with foil.