Place the butter, shortening and applejack into the refrigerator for 1 hour.
In the bowl of a food processor, combine the flour, salt and sugar by pulsing 3 to 4 times. Add the butter and pulse 5 to 6 times until the texture looks mealy. Add the shortening and pulse another 3 to 4 times until incorporated.
Remove the lid of the food processor and sprinkle in 5 tablespoons of the applejack. Replace the lid and pulse 5 times. Add more applejack as needed, and pulse again until the mixture holds together when squeezed. Weigh the dough and divide in half. Shape each half into a disk, wrap in plastic wrap and refrigerate for at least 1 hour and up to overnight.
For the filling:
Peel and core the apples. Slice into 1/2-inch thick wedges. Toss all of the apples with 1/4 cup of the sugar, place in a colander set over a large bowl and allow to drain for 1 1/2 hours.
Transfer the drained liquid to a small saucepan, place over medium heat and reduce to 2 tablespoons. Set aside to cool. Toss the apples with the remaining sugar, tapioca flour, jelly, cider, lime juice, salt and grains of paradise.