Place the chicken halves in a bowl. Pour buttermilk over the chicken, add salt, pepper, and paprika. Cover and refrigerate for at least 2 hours, or up to 12 hours.
Prepare the dry dredge: In a shallow dish, whisk together the flour, cornstarch, and baking powder.
Prepare the wet batter: In a separate bowl, whisk the club soda into the dry dredge mixture until just combined. Do not overmix; lumps are fine.
Remove chicken from buttermilk, letting excess drip off. Dip the chicken into the dry dredge, coating completely. Shake off excess flour.
Dip the floured chicken back into the wet batter, ensuring it is fully coated.
Immediately return the chicken to the dry dredge. Press the breading firmly onto the chicken to create craggy, thick pieces. Set aside on a wire rack while you heat the oil.
Heat 2 inches of vegetable oil in a deep skillet or Dutch oven to 325 degrees Fahrenheit.
Carefully place chicken into the hot oil, ensuring not to overcrowd the pan. Fry for 6 to 8 minutes, turning occasionally, until golden brown and cooked through (internal temperature of 165 degrees Fahrenheit).
Remove chicken and place on a clean wire rack set over a baking sheet. Immediately place one slice of Gouda cheese on top of each hot patty. Let rest for 2 minutes to allow the cheese to melt.
While the chicken rests, lightly toast the brioche buns.
Mix the mayonnaise, hot sauce, and relish to create the spicy sauce.
Assemble the burgers: Spread the spicy sauce on both halves of the toasted buns. Place shredded lettuce and pickle chips on the bottom bun. Top with the cheesy chicken patty and the top bun.
Notes
For extra crispiness, you can double-dip the chicken in the wet batter and dry dredge one more time before frying.
If you prefer baking, place the breaded chicken on a rack over a baking sheet and bake at 400 degrees Fahrenheit for 20-25 minutes, flipping halfway, then add cheese for the last 2 minutes.
Use a meat thermometer to confirm the chicken reaches 165 degrees Fahrenheit internally to guarantee safety and juiciness.