Special equipment: Special equipment: 9 1/2-inch deep-dish pie plate
For the crust: Put the flour, sugar and salt in the bowl of a food processor. Pulse a couple of times to mix. Add the shortening and pulse to combine. Add the butter and pulse until the mixture is coarse and a few pieces of butter are still visible. Add the cheese and pulse just to combine. Add the apple cider vinegar and ice water. Pulse until the dough just begins to come together; it should look moist but still crumbly and not yet a complete ball. Turn the dough out onto a piece of plastic wrap and quickly form into a ball. Divide into 2 disks, one slightly larger than the other, wrap in plastic and refrigerate 1 hour or overnight. The less you work the dough the flakier it will be.
For the filling: Melt the butter in a large skillet over medium heat. Once melted, add the apples (if the apples cannot sit in one even layer, use 2 medium saute pans to cook the apples in batches). Add the sugar, lemon zest, lemon juice, cinnamon, nutmeg and salt. Continue cooking over medium heat until the apples release their liquid and it begins to boil, 12 to 15 minutes. The apples will be tender but will not lose their shape. Once the apples are tender, add the flour and cook until the liquid has thickened and coats the back of a spoon, about 1 minute longer. Remove the apples to a baking sheet and spread out in an even layer to cool to room temperature, about 1 hour.
Preheat the oven to 425 degrees F. Place a baking sheet in the oven while preheating so it will be hot by the time the pie goes into the oven.
Roll the larger disk of dough on a lightly floured surface to about 12 inches in diameter and 1/8-inch thick, to fit a 9 1/2-inch deep-dish pie plate. Roll the dough onto your rolling pin and then unroll it over your pie plate. Trim the excess dough to 1/2-inch, and do not crimp the edges. Place the cooled filling in the pie plate, mounding at the center.
Roll the smaller disk of dough on a lightly floured surface to 1/8-inch thick and 10 inches in diameter and place on top of the filling. Fold the edges under, and decoratively crimp the edges. Cut slits in the top crust to allow steam to escape. Whisk the egg with 1 teaspoon water and brush the top of the pie.
Place the pie on the preheated baking sheet and bake for 10 minutes. Decrease the temperature to 350 degrees F and continue baking until the top crust is golden brown and crisp to the touch and the bottom crust is browned, 50 to 60 minutes. Cool to room temperature and serve.
Use a cave-aged Cheddar for a stronger cheddar taste.