To make the crust: In a large mixing bowl, whisk together the flours, sugar, and salt; set aside
In a large measuring cup or a small bowl, beat the egg with 1/3 cup of the water and the vinegar.
Add the shortening and butter to the flour and, using a pastry blender, cut it in until the butter resembles small peas. Add the egg mixture, gently tossing and mixing with your hands or a fork just until the dough comes together in a ball. If the dough seems too dry, add a little more ice water, about 1 tablespoon at a time as necessary.
Gather the dough together on a lightly floured work surface and divide it evenly into 2 balls. Wrap each half in plastic wrap and flatten with the palm of your hand into a disk. Chill for at least 1 hour. The dough can be stored in the refrigerator for up to 4 days or in the freezer for up to 2 weeks.
To roll out the dough: Dust your hands and your rolling pin with flour. On a lightly floured piece of parchment, roll out the chilled dough into a 12-inch round. Roll the dough from the center out, rotating it slightly as you roll to prevent sticking and to keep it round. Dust off any excess flour using a dry pastry brush. Roll the dough onto the rolling pin and place it in the pie dish. Carefully arrange the dough to slump inside the dish and press it into the edges. Trim the excess dough with kitchen shears, leaving about a 1-inch overhang. Wrap and place the pie dish in the refrigerator to chill the dough for at least 30 minutes.
Roll out the remaining disk of dough on parchment and transfer it to a baking sheet. Wrap in plastic wrap and return the dough to the refrigerator to chill for at least 30 minutes.
When ready to bake, position a rack in the lower third of the oven and preheat the oven to 400°F.
To make the caramel: Combine the granulated sugar, fleur de sel, and 1/2 cup water in a medium saucepan and cook over medium-high heat, stirring constantly, until the sugar is dissolved. Wash down any crystals from the sides of the pan with a wet pastry brush. Turn the heat down to medium and cook, without stirring, until the syrup becomes a medium-dark amber caramel, about 15 minutes; you can carefully swirl the pan around to check the color.
Carefully remove the pan from the heat and immediately stir in the butter and heavy cream. Be careful of hot steam when the heavy cream is added, and do not worry if the butter starts to separate--it will come together once the caramel is cooled.
Transfer the caramel to the bowl of a stand mixer fitted with the whisk attachment (or to a medium mixing bowl, if using a handheld mixer) and beat on low speed until the caramel cools and starts to come together. Set aside.
To make the filling: Put the lemon zest and lemon juice in a large bowl. Add the apples and toss gently.
In a small mixing bowl, whisk together the flour, cardamom, allspice, nutmeg, cinnamon, and turbinado sugar. Gently toss the apples with this mixture.
To assemble the pie: Place the bottom piecrust in a 9-inch pie pan and prick the bottom of the crust gently all over with a fork. Layer the apple mixture in the crust, making sure there are no gaps between the apples. Pour 1/4 cup of the caramel mixture on top of the apples. Reserve the remaining caramel for serving.
Place the top piecrust on a work surface and cut 4 to 8 vent holes in the center with a mini cookie cutter in whatever shape you like. Reserve the cutout pieces for decorating the crust. Brush the rim of the bottom piecrust with the egg wash to create a seal. Place the top crust over the pie filling and seal and crimp the edges, trimming the excess dough. Decorate with the cutout pieces, and brush the entire crust with the egg wash. Sprinkle lightly with turbinado sugar and a pinch of fleur de sel.
Place on the prepared baking sheet and bake for 20 minutes. Turn the oven down to 375°F and bake for an additional 45 to 60 minutes, until the caramel blossoms into big, thick, syrupy bubbles and the crust turns golden brown; test the apples with a small paring knife to make sure they are tender but not mushy. Remove from the oven and let cool for at least 4 hours before serving to allow the juices and caramel to thicken. The pie is best served the same day, but it can be covered with plastic wrap and refrigerated for up to 3 days.
To serve, slice the pie into wedges and top with a drizzle of the caramel sauce and, if desired, a scoop of ice cream.
Excerpted from The Back in the Day Bakery Cookbook by Cheryl Day & Griffith Day/Artisan Books, 2012.