In a medium mixing bowl, whisk together the flour, sugar, baking powder, salt, and cayenne pepper.

Using the large holes of a box grater, grate the cold butter directly into the flour mixture. Using a pastry blender or fork, cut in the shortening until the mixture resembles coarse crumbs.

Stir in the shredded cheddar cheese until evenly combined.

Gradually add the ice water, a few tablespoons at a time, stirring just until the dough begins to pull away from the sides of the bowl and holds together when pressed. You may not need the entire ⅔ cup. Do not overmix.

Divide the dough in half. Shape each half into a flat disk, wrap each disk in plastic wrap, and refrigerate for at least 1 hour.

Preheat the oven to 450°F.

Remove one disk of dough from the refrigerator. On a lightly floured work surface, roll the dough into a circle about 13 inches in diameter and ⅛ inch thick.

Carefully transfer the dough to a 9-inch pie plate. Gently press the dough into the bottom and sides of the plate. Trim any excess dough, leaving enough around the edge to seal with the top crust later.

Line the crust with parchment paper and fill with pie weights, dried beans, or uncooked rice. Bake for 8 minutes.

Carefully remove the parchment paper and pie weights. Return the crust to the oven and bake for an additional 3 minutes. Remove from the oven and allow the crust to cool slightly.

While the crust is cooling, combine the sliced apples, sugar, flour, lemon juice, cinnamon, and nutmeg in a large mixing bowl. Toss until the apples are evenly coated.

Spoon the apple filling into the prebaked pie shell, spreading the apples into an even layer.

Remove the remaining disk of pie dough from the refrigerator. On a lightly floured work surface, roll the dough out to approximately ⅛ inch thick.

Carefully drape the dough over the apple filling. Trim any excess dough, then seal and crimp the edges of the top and bottom crusts together.

Cut several slits in the top crust to allow steam to escape.

Place the pie on a rimmed baking sheet to catch any drips. Bake at 450°F for 20 minutes.

Reduce the oven temperature to 350°F. Check the crust at this point. The cheddar cheese crust tends to brown more quickly than a traditional pie crust. If it is already nicely browned, loosely tent the entire pie with aluminum foil or cover the edges with a pie shield.

Continue baking for an additional 20–35 minutes, checking the crust periodically and covering it as needed to prevent overbrowning. The pie is done when the crust is golden brown, the apples are tender when pierced through one of the vents with a thin knife, and the filling is bubbling through the vents.

Transfer the pie to a wire rack and allow it to cool before slicing and serving.