Make the crust. Place 3/4 of the butter in the freezer for about 30 min, and keep the rest in the fridge. In a food processor combine flour, salt, and sugar. Add refrigerated butter, pulse about 10 times. Add frozen butter, pulse again till the mixture resembles coarse meal, with some blueberry-size crumbs.
Add ice water, and immediately pulse until water is incorporated, about 10 more times. lay out 2 pieces of plastic wrap, separate the dough in halves, and place each half on the plastic wraps. Bring the edges of wraps together to gather dough. Press into disks. Roll each disk (still in plastic) into 1/2-inch thick rounds. Refrigerate for at least 45 min.
Make the filling. In a medium sauce pan mix sugar and water and heat until the sugar is dissolved. Add butter and bring to a slow boil. Continue cooking until the mixture turns golden brown. Remove it from heat immediately and add cream, whisk the final mixture on low heat , sprinkle the sea salt, and let cool while preparing the apples.
Core, peel and cut apples using a mandoline. Sprinkle with juice, sugar, and spices, and toss well. Preheat the oven to 375°F/190°C.
Assemble the pie. Roll each piece of the dough into a disk that can fit a 9-inch pie-pan. Line the pan with the first disk. Lay 1/3 of the apple mixture on the bottom and pour a layer of 1/3 of prepared caramel. Repeat the layering two more times.
Cover the pie with the second disk, cutting it through for the air to escape any way you like, or cut the second disk into strips and make a lattice top like mine, pinch the edges together.
Brush with the beaten egg and sprinkle with sugar and salt flakes.
Bake. Place the pie on a baking sheet and bake for 20 minutes at 375°F/190°C. Reduce the heat to 350°F/175°C and bake for another 45 minutes.
Serve warm or cold with ice cream.