Special equipment: a deep-frying thermometer
Combine the pickle juice, 1 tablespoon hot sauce and 1/2 teaspoon salt in a large resealable plastic bag. Add the chicken thighs, seal the bag and shake to evenly disperse. Refrigerate and let marinate for at least 2 hours or up to overnight. When ready to fry, remove the chicken from the marinade and pat dry. (Discard the marinade.)
Fill a large, wide Dutch oven or pot with about 2 1/2 inches of oil, then add the shortening. Attach a deep-frying thermometer to the side of the pot and bring to 350 degrees F over medium heat. Line a large baking sheet with paper towels and set aside.
Whisk together the flour, paprika, mustard, cayenne, 1/2 teaspoon salt and 3/4 teaspoon pepper in a large bowl. Pour in 1/4 cup of buttermilk and use your fingers to work it in, making the flour shaggy.
Whisk the remaining cup of buttermilk and 1 tablespoon hot sauce together in another large bowl.
Dip one of the chicken thighs in the buttermilk mixture and let the excess drip off. Add to the flour mixture, making sure to press the chicken into the flour to fully coat. Set on a plate. Repeat with as many chicken thighs as your pot can hold comfortably in one batch. (As the chicken sits, if you notice it getting soggy in any spots, then dredge it in the flour mixture again.)
Once the oil is at 350 F, being careful of the hot oil, slide the breaded chicken into the oil, turn off the heat and allow the temperature to drop all the way to 300 degrees F. When the temperature reaches 300 degrees F, turn the heat back on low and continue to fry, turning the chicken as needed for even browning, while adjusting the heat to maintain a temperature around 300 degrees F. Fry until the chicken is a deep golden brown with an internal temperature of about 170 degrees F, about 5 minutes. Remove to the lined baking sheet and sprinkle with a little salt. If you're working in batches, dredge the remaining chicken and increase the heat to 350 degrees F again and repeat the process.
Once all the chicken is fried, split the buns and toast in a toaster oven or broiler, cut-side up, until golden. Put 4 pickles on each bottom bun, add a piece of fried chicken and sandwich with the top buns. Serve hot or wrap in foil to transport.
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