Pound the chicken halves between plastic wrap until they are about 1/2 inch thick.
In a bowl, mix the buttermilk and hot sauce. Soak the chicken pieces in this mixture for at least 30 minutes, or up to 4 hours in the refrigerator.
In a shallow dish, whisk together the flour, cornstarch, salt, pepper, paprika, garlic powder, and cayenne pepper. This is your dry dredge.
Remove one piece of chicken from the buttermilk, letting excess drip off. Dredge it thoroughly in the flour mixture, pressing firmly to coat.
Dip the floured chicken piece quickly back into the beaten egg, then immediately return it to the flour mixture. Press and shake to create a thick, craggy coating. Set the coated chicken aside on a wire rack. Repeat for all pieces.
Heat about 2 inches of vegetable oil in a deep skillet or Dutch oven to 350°F (175°C). Use a thermometer to monitor the temperature.
Carefully place two chicken pieces into the hot oil, ensuring you do not overcrowd the pan. Fry for 4 to 6 minutes per side, until golden brown and cooked through (internal temperature reaches 165°F or 74°C).
Remove the cooked chicken and place it on a clean wire rack set over a baking sheet to drain excess oil.
While the chicken is hot, immediately place one slice of cheese on top of each fillet. Cover the rack loosely with foil for 1 minute to trap heat and melt the cheese.
Lightly toast the hamburger buns.
Spread mayonnaise on the bottom and top halves of each bun.
Assemble the burgers: place the cheesy chicken fillet on the bottom bun, top with desired lettuce, tomato, and pickles, then place the top bun on. Serve immediately.
Notes
For extra crispiness, use a double-dredge technique: flour, then egg, then flour again, then egg again, then a final coating of flour.
Maintain the oil temperature between 325°F and 350°F. If the oil is too cool, the coating will absorb too much oil and become soggy.
If you prefer not to deep fry, you can air fry the coated chicken at 380°F (195°C) for 15-20 minutes, flipping halfway, until golden and cooked through.