Make the aioli. In a small bowl, stir together the mayonnaise, chopped pickle, pickle brine, garlic, and mustard. Season with salt and pepper. Cover and refrigerate for at least 30 minutes or up to overnight.

Make the fried chicken. Fill a large deep skillet or pot with about 3 inches of oil. Clip a deep-fry thermometer to the side of the pan and heat the oil over medium-high heat until it reaches 350°F. Set a wire rack over a baking sheet or line a plate with paper towels.

In a shallow dish, whisk together 1 cup of the flour, the pickle brine, paprika, garlic powder, salt, black pepper, and cayenne until the batter is smooth. Place the remaining 1 cup flour in a separate shallow dish.

Working with one piece at a time, dip each chicken thigh in the flour to coat, shaking off any excess, then dip it into the batter, turning to coat and letting any excess drip off. Carefully place the battered chicken in the hot oil and repeat with another piece. Cook until the chicken is golden brown and cooked through-it should register 165°F on an instant-read thermometer-5 to 8 minutes per side. Transfer the fried chicken to the rack to drain. Repeat to batter and fry the remaining chicken thighs, allowing the oil to return to temperature between batches.

Assemble the sandwiches. Melt the butter in a large cast-iron skillet over medium-high heat, tilting the pan to coat the bottom. Add the buns cut-side down and toast until golden brown, about 3 minutes. Transfer to a plate.

Spread a layer of the pickle aioli over each bottom bun. Place a fried chicken thigh on top and finish with dill pickles, lettuce, tomato, and onion. Serve while crispy and hot!