Combine the cassava flour, tigernut four, arrowroot flour, sugar, and salt in a large bowl. Mix well with a whisk.
Take the palm shortening out of the fridge and cut up in small chunks using a sharp knife.
Add to the flour and continue working it into the flour with a pastry knife until the palm shortening is broken down into pea-sized pieces and coated with flour. You may use a sharp knife if you don’t have a pastry cutter. Stir in the ice-cold water and mix the dough with a rubber spatula until you obtain a rough and chunky dough. Do not overwork the dough; it is supposed to look and feel rough on purpose.
Divide the dough into 3 even portions, form into balls using your hands and refrigerate while you prepare the persimmons.
Cut the stem out of the Fuyu persimmons and then cut them in half. Cut each half into ¼-inch thick slices using a serrated knife.
Preheat the oven to 400°F and place the rack in the middle. Line a baking sheet with parchment paper.
Cut a large piece of parchment paper and place it on your work area (countertop or wooden board). Lightly dust with arrowroot powder. Place one ball of dough in the center and dust with arrowroot as well. Place another piece of parchment paper over the dough and flatten a bit with your hand. Using a rolling pin, continue to flatten the dough carefully until you obtain a disc of about ⅛-inch thick.
Spread 2 tbsp of applesauce in the center of the disc (leaving approximately ¾-inch space on the edges). Top with slices of persimmon.
Fold up the edges around the fruit, pressing on the pleats to stick them together. Transfer the galette on the parchment-lined baking sheet using a large spatula. Repeat with the remaining balls of dough.
Bake in the oven until the crust turns golden brown, 35 to 40 minutes, rotating the baking sheet halfway through. Enjoy warm right out of the oven!