Start by making your dough. Put the flour, sugar and salt in the processor bowl of a food processor and pulse once or twice to combine. Add the butter and continue to pulse until the mixture resembles coarse crumbs. Slowly add ¼ cup of water as you continue to pulse, adding more water a tablespoon at a time if necessary; stop pulsing when the dough just starts to come together. Remove the dough from the food processor and form into 2 disks. Wrap each one in plastic wrap and refrigerate for at least 1 hour, or up to 3 days.
Remove one disk of dough from the refrigerator and place on a lightly floured surface. Roll the dough out into a 13-inch circle. Transfer the dough to a 9-inch pie plate, and gently press the dough into the bottom of the pan and against the sides. Leave the overhanging dough in place and refrigerate for at least 30 minutes, or until completely chilled and firm.
Preheat oven to 375°F, then prepare your pie filling. Core, peel, and thinly slice apples. Combine with lemon juice and granulated sugar, and set aside to macerate for 20-30 minutes. In a large bowl, stir together demerara sugar, spices, and bitters. Remove apple slices from soaking liquid and add to sugar mixture, toss to coat evenly.
Place half the apple filling into prepared bottom crust. Pour ¼ cup of the caramel over top and sprinkle with ½ teaspoon of flaky salt. Follow that with remaining apple filling and then top with more caramel, ¼ to ½ cup depending on desired sweetness. Sprinkle remaining salt evenly over pie filling.
Roll out remaining pie dough and cut into strips. Place in a lattice pattern over apples. Crimp edges of crust. Brush with beaten egg and lightly sprinkle with additional demerara sugar. Place pie on a preheated baking sheet and bake 20 minutes. Reduce oven temperature to 325F and continue baking 30 to 45 minutes. Test doneness of apples with a long toothpick or small knife. They should be just soft. Let cool at least 1 hour before slicing.
*Heavily adapted from the Four & Twenty Blackbirds recipe