For the crust
2 1/2 cups all purpose flour
1 tablespoon sugar
3/4 teaspoon salt
10 tablespoons (1 1/4 sticks) chilled unsalted butter, cut into 1/2-inch pieces
1/3 cup chilled solid vegetable shortening, diced
6 tablespoons (or more) ice water
For the filling
6 large apples (I use pink lady apples), peeled, cored and sliced
1/3 cup granulated sugar, plus more for sprinkling (optional)
2 tablespoons flour
1 egg, beaten
For the caramel (makes 1 cup)
1/2 cup heavy cream
2 1/2 tablespoons unsalted butter
1 teaspoon fleur de sel, plus more for sprinkling (optional)
3/4 cup granulated sugar
2 tablespoons light corn syrup
2 tablespoons water
Blend flour, sugar and salt in processor. Add butter and shortening and cut in using on/off turns until mixture resembles coarse meal. Add 6 tablespoons ice water and process until moist clumps form, adding more water by teaspoonfuls if dough is dry. Gather into ball; divide into 2 pieces. Flatten each into disk. Wrap each in plastic; chill 2 hours.
Preheat the oven to 375F. On a floured work surface, roll dough into two circles with a diameter slightly larger than 9-inches. Press one of the circles into a pie plate. Place the other one on parchment or wax paper and cover with plastic wrap. Place both in the refrigerator to chill until ready to use.
In a large bowl, toss apple slices with sugar and flour. Set aside.
Make the caramel: bring cream, butter and fleur de sel to a boil in a small saucepan, then remove from heat and set aside. In a separate heavy saucepan, bring sugar, corn syrup and water to a boil, stirring until sugar is dissolved.
Boil, without stirring but gently swirling pan, until mixture is a light golden caramel color. Carefully stir in cream mixture (mixture will bubble up) and simmer, stirring frequently, until caramel registers 250F on thermometer. Take off heat, pour into a tall glass or ceramic container and set aside.
Drain any liquid that may have accumulated in the apple bowl. Remove pie plate with crust from the refrigerator. Add about a quarter of the prepped apples and drizzle about a quarter of the caramel sauce on top. (Note: if the caramel has set to the point where it will not pour, place the glass or ceramic container in the microwave for just a few seconds or place in a hot water bath to liquify the caramel.) Repeat three more times, adding a quarter of the apples and a quarter of the caramel each time, until all of both are done. Be sure to finish with the caramel.
Remove the top crust from the refrigerator and use it to cover the pie. Crimp around the edges and make several slits or cut out holes for venting. Lightly brush the top of the crust with egg and sprinkle with granulated sugar and, if you like your sweet-and-salty treats on the saltier side, just a little fleur de sel.
Line a baking sheet large enough to hold the pie plate with aluminum foil. Place pie plate on the sheet and bake uncovered until the crust turns a light golden brown, about 35 minutes. Cover loosely with tin foil and continue baking until the pie is bubbling and the crust is the perfect golden brown, another 15-25 minutes. Allow to cool before serving.