Make the Dough: Process 1½ cups of the flour, the sugar and salt together in a food processor until combined, about 2 one-second pulses. Add the butter and shortening and process until a homogenous dough just starts to collect in uneven clumps, about 7 to 10 seconds (the dough will resemble cottage cheese curds with some very small pieces of butter remaining, but there should be no uncoated flour). Scrape down the sides and bottom of the bowl with a rubber spatula and redistribute the dough evenly around the bowl. Add the remaining 1 cup flour and pulse until the mixture is evenly distributed around the bowl and the mass of dough has been broken up, 4 to 6 quick pulses. Empty the mixture into a medium bowl.
Sprinkle the vodka and water over the mixture. With a rubber spatula, use a folding motion to mix, pressing down on the dough until it is slightly tacky and sticks together.
Divide the dough into two portions. Roll one out on a generously floured work surface to a 12-inch circle. Transfer to a 9-inch pie plate, leaving at least a 1-inch overhang on each side. Working around the circumference, ease the dough into the plate by gently lifting edge of the dough with one hand while pressing into the plate bottom with other hand. Dust off any excess flour with a pastry brush. With scissors or kitchen shears, trim the edge so that it overhangs by 1 inch.
For solid top crust, roll remaining dough as before (into a 12-inch circle); for a lattice-top pie, roll it into a 9- by 15-inch rectangle instead. Transfer to a parchment-lined baking sheet. Wrap both portions in plastic wrap and refrigerate for at least 2 hours and up to one day in advance.
Prepare the Filling: In a large bowl, toss the apple slices with the lemon juice, then drain in a colander. In a medium bowl, whisk together the sugar, flour, cinnamon, allspice and nutmeg. Return the apple slices to the large bowl, add the sugar mixture and toss to combine.
Assemble the Pie: Remove the dough-lined pie plate from the refrigerator. Use a slotted spoon to remove one-third of the apple mixture from the bowl and layer in the bottom of the crust so there are hardly any gaps. Top with one-third of the salted caramel sauce, then repeat twice more with layers of apples and salted caramel sauce.
If cutting shapes out of the reserved dough round, do so now. If using a lattice-top, cut the rectangle of dough into 1-inch strips.Top the pie with either the dough round (or lattice-top) and trim the edges of the dough a ½-inch over the edges of the pie plate. Crimp the edges of the pie and then place in the refrigerator for 30 minutes before baking.
Bake the Pie: While the pie is chilling, place a rimmed baking sheet on the center rack of the oven and preheat to 350 degrees F.
When ready to bake, if you have not cut out shapes of the dough round or done a lattice-top, use a sharp knife to cut slits in the top crust. Brush the top of the pie with the egg wash and sprinkle with the sugar. Place the pie on the baking sheet in the oven and bake for 30 minutes, or until the edges of the crust start to brown. Remove the pie from the oven and cover the edges of the crust with a pie ring.
Continue to bake for an additional 25 to 30 minutes, until the crust is golden brown. Remove the pie ring and continue to bake for an additional 5 minutes if the edges of the crust are a little pale compared to the rest of the crust. Transfer the pie to a wire rack and allow to cool completely before serving. The pie can be kept, covered, at room temperature or in the refrigerator for up to 2 days.
Crust: This is my very favorite pie crust, but if you prefer not to use shortening, give my all-butter pie crust a try.
Apples: Any combination of tart/crisp apples will work in this pie. Some suggestions: Granny Smith, Jonagold, Pink Lady, Fuji, Honeycrisp, and Braeburn.
Salted Caramel Sauce: You can substitute 1 cup of store-bought caramel sauce.
Storage: The baked and cooled pie can be stored at room temperature or in the refrigerator for up to 2 days.
Freezing Unbaked Pie: Place the fully assembled, unbaked pie on a baking sheet and freeze until solid. Wrap in a few layers of plastic wrap, then a layer of aluminum foil, and freeze for up to 2 months. Unwrap and bake directly from frozen. It will take more time to bake.
Freezing Fully Baked Pie: Bake the pie, then cool completely. Wrap in a few layers of plastic wrap, then aluminum foil, and freeze for up to 2 months. Thaw in the refrigerator overnight. Unwrap and rewarm in a 350-degree oven for 15 to 20 minutes.