Refrigerate the butter, shortening, and applejack for 1 hour.
Combine the flour, salt, and sugar in your food processor by pulsing 3 to 4 times. Add the butter and pulse until the texture looks mealy, 5 to 6 pulses. Add the shortening and pulse until incorporated, another 3 to 4 pulses.
Pop off the lid and sprinkle in 5 tablespoons of the applejack. Replace the lid and pulse 5 times. Add more applejack as needed until the mixture holds together when squeezed. Weigh the dough and divide it in half. Shape each half into a disk, wrap in plastic wrap, and refrigerate for at least 1 hour or up to overnight.
Peel and core the apples and slice into 1/2-inch thick wedges. Toss apples with 1/4 cup of the sugar, place in a colander set over a large bowl, and drain for 1 1/2 hours.
Transfer the drained liquid to a small saucepan, place over medium heat, and cook until it is reduced down to a mere 2 tablespoons, then set aside to cool. Toss the apples with the remaining sugar, the tapioca flour, jelly, cider, lime juice, salt, and grains of paradise.
Heat oven to 425ºF. Line a half-sheet pan with parchment paper.
Remove one disk of dough from the refrigerator. Place the dough on a lightly floured piece of waxed paper or parchment paper, lightly sprinkle the top with flour, and roll out into a 12-inch circle. Place in a 9 1/2- to 10-inch tart pan that is 2 inches deep. Gently press the dough into the side of the pan, crimping and trimming the edges as necessary. Set a pie bird in the center of the bottom of the pan to help vent steam out of the pie. If you don't have one, you can skip it, but your pie won't be quite as solid inside.
Arrange the apples in the unbaked pie shell in concentric circles starting around the edges, working toward the middle and forming a slight mound in the center of the pie. Pour any liquid that remains in the bowl over the apples.
Roll out the second disk of dough as the first. Place this dough over the apples, pressing the pie bird through the top crust. Press the edges of the dough together around the rim of the pie. Brush the top crust with the reduced juice, but avoid the rim. Trim off any excess dough.
Place the pie on the prepared half-sheet plan and bake on the floor of the oven for 30 minutes. Transfer to the lower rack of the oven and continue to bake until the apples are cooked through by not mushy, about 20 minutes. Remove to a wire rack and cool at least 4 hours before serving.