Cut each chicken breast roughly in half and pound until it reaches your desired thickness.
Whisk together the pepper, gochujang, soy sauce, sherry, and garlic in a medium-sized bowl.
Add the chicken, cover, and marinate for at least 1 hour and up to 4 hours.
Prepare the slaw: Roughly chop the kimchi into bite-sized pieces.
Whisk together the mayo, rice wine vinegar, honey, ground black pepper, and sesame seeds in a bowl.
Add the kimchi, cabbage slaw mix, and chopped scallions and toss until well coated. Set aside.
Prepare the breading: Whisk together the cornstarch, baking powder, salt, and pepper in a shallow dish.
Heat the vegetable oil in a heavy-bottomed, large pan until it reaches 325 F.
Dip the chicken in the breading until all of the sides are well coated.
Add the chicken to the hot oil and cook for 5 minutes on each side, then continue cooking until golden brown and the center reaches 165 F.
Remove from the oil and drain on a paper towel–lined plate.
Prepare the sauce: In a small saucepan, combine the sesame oil, gochujang, brown sugar, soy sauce, garlic, and ketchup. Bring to a simmer and cook on low until thickened, about 3 minutes.
To make the sandwich, spread both sides of the buns with butter.
Toast the buns in a pan or griddle until golden brown.
Dip the fried chicken into the sauce and coat all of the sides.
Add the chicken to the bottom bun and top with the slaw.