Pulse the flour, sugar, and salt in a food processor until combined. Add the cubed shortening and pulse until pea-sized crumbs form. Transfer the mixture to a large bowl.

Whisk the beaten egg, vinegar, and water in a small bowl. Pour over the flour mixture and stir until a sticky dough comes together. Cover and refrigerate for at least 1 hour.

Divide the chilled dough in half and let it rest at room temperature for about 15 minutes while you prepare the filling.

Preheat the oven to 425°F and place a rack in the lowest position.

Add the sliced apples to a large bowl. In a separate bowl, stir together the sugar, flour, cinnamon, and salt. Toss the mixture with the apples, then stir in the vanilla. Set aside.

On a lightly floured surface, roll each dough half into a 12-inch circle. Fit one circle into a 9-inch deep-dish pie plate.

Spoon the apple filling into the crust and scatter the butter pieces over the top.

Place the second crust over the filling. Press and crimp the edges to seal, then cut a few small slits in the top for steam to escape.

Whisk the egg and water together and brush the mixture over the top crust.

Bake for 15 minutes. Reduce the oven temperature to 350°F, loosely cover the pie with foil, and bake for 45 to 50 minutes more, until the crust is golden and the apples are tender.

Cool the pie for at least 1 hour before slicing.