Make the crust: Pulse the flour, sugar and salt in a food processor. Add the shortening and pulse until the mixture looks like cornmeal. Add the butter and pulse until it is in pea-size pieces. Add the vinegar and 5 to 6 tablespoons ice water; pulse until the dough just starts forming a ball. Divide between two pieces of plastic wrap. Shape each into a disk and wrap. Refrigerate until firm, at least 1 hour or overnight.
Make the filling: Peel the apples and slice 1/4 inch thick; toss with 2/3 cup sugar, the lemon juice and salt in a large bowl. Melt 2 tablespoons butter in a large skillet over medium-high heat. Add half of the apples; cook, stirring, until slightly softened, 5 minutes. Sprinkle with 1 tablespoon flour. Cook, stirring, until the juices thicken, about 1 minute. Transfer to a separate large bowl. Repeat with the remaining butter, apples and flour; add to the bowl with the cooked apples. Gently fold in the sour cream and vanilla paste. Let cool completely.
Roll out 1 disk of dough into a 12-inch round on a lightly floured surface. Ease into a 9 1/2 -inch deep-dish pie plate. Add the filling, slightly mounding it in the center. Refrigerate until ready to bake.
Put a baking sheet on the lowest oven rack and preheat to 425 degrees F for 30 minutes. Meanwhile, roll out the remaining disk of dough into a 12-inch round on a floured surface. Lay the dough over the filling and press the crusts together around the edge; fold the overhanging dough under itself. Crimp with your fingers or a fork. Refrigerate until firm, 30 minutes.
Combine the remaining 2 tablespoons sugar and the apple pie spice. Brush the top crust with the beaten egg and sprinkle with the spiced sugar. Make a few slits in the crust with a knife to let steam escape. Transfer the pie to the hot baking sheet and bake 15 minutes. Reduce the oven temperature to 375 degrees F and continue baking until the crust is golden and the filling is bubbling, rotating the pie as needed, 1 hour 15 minutes. (Tent with foil if the crust browns too quickly.) Transfer to a rack and let cool completely.