Make the sauce. Whisk mayonnaise, relish, mustard, vinegar, sugar, garlic powder, onion powder, and paprika until smooth. Cover and chill at least 30 minutes.
Prep the chicken. Pound chicken to ¼ inch thick and trim into 8 small bun-sized rounds. Soak in buttermilk and hot sauce for 30 minutes (or overnight).
Bread it. Stir flour, baking powder, paprika, garlic powder, onion powder, salt, and pepper together. Dredge each piece, dip back into the buttermilk, then dredge a second time, pressing to build a craggy coat.
Fry. Heat 1 inch of oil to 350°F (175°C). Fry patties 3–4 minutes per side until deep golden and 165°F (74°C) inside. Drain on a wire rack.
Toast the buns. Lightly toast all three bun layers per sandwich (use the sliced extra buns for the middle "club" layer).
Assemble. Bottom bun → sauce → onion → lettuce → cheese → chicken patty → middle bun → sauce → onion → lettuce → pickles → second chicken patty → top bun. Serve immediately.