To make the crust: Frigidity is all-important here. Place butter and shortening in freezer for at least 1 hour before mixing. Pulse flour, salt and cayenne pepper in food processor. Remove the lid and tuck the butter and shortening into the mixture. Pulse the machine 4 times to cut into the butter. Add 1 1/3 cups of cheese and pulse 4 more times. Sprinkle half of the water over the flour mixture and pulse 5 or 6 times. Add the rest of the water and pulse 5 or 6 more times, until the pastry looks like very coarse crumbs.
Move mixture to a chilled bowl and work until dough is formed. Round into 2 balls, one slightly larger than the other. Wrap each dough ball in 2 layers of plastic wrap and press them into disks. Chill at least 3 hours.
Preheat oven to 400 degrees F
To make the filling. Peel and slice the apples. In a bowl, mix the apples, lemon juice, dark brown sugar, flour, cinnamon, vanilla, salt, and nutmeg with your hands. Set aside and retrieve the pie crusts from the refrigerator.
To finish: Roll the dough into 2 circles that are 2 to 3 inches wider in diameter than your pie shell or plate. Press the bottom crust into the pie plate and mound apples until they fill the crust and dome slightly. Scatter the butter over the mound. Cut a center vent the size of a dime into the top crust. Sprinkle the remaining cup of shredded cheese onto the crust. With a rolling pin, roll lightly to press the cheese into the dough. Roll the crust onto the rolling pin so that the rolling pin resembles the sausage in a pig-in-a-blanket. Unroll the crust, cheese-side down, on top of the apples. Bake at 400 degrees F, for 10 minutes. Reduce the heat to 350 degrees F and bake for 40 to 50 minutes longer. Crimp the edges of the crust.
Recipe courtesy of John T. Edge, Apple Pie: An American Story, Putnam 2004