Drain the juice from the jars of pickles into a baking dish or bowl. Add the chicken and flip to coat. Let brine, refrigerated, for at least 1 hour and up to 4 hours.
Preheat the oven to 300 degrees F and line a baking sheet with a cooling rack. Fill a Dutch oven halfway with oil. Heat the oil over medium-high heat to 320 degrees F.
In 3 bowls or baking dishes, set up a breading station. In the first dish, mix the flour with the salt, paprika and pepper. In the second dish, mix the hot sauce, mustard and eggs. Put the cracker crumbs in the third dish.
Dredge each chicken piece first in the flour mixture, then in the egg mixture, then in the crackers. Place the breaded chicken pieces in the oil a few at a time. Don't overcrowd the pan; the pieces should not be touching in the oil. Fry, turning occasionally, until the chicken is a deep golden brown and fully cooked to 165 degrees F when checked with a thermometer, 5 to 6 minutes per side. Transfer to a baking sheet and place in the oven until ready to serve.