In a large bowl, whisk together soy sauce, brown sugar, honey, gochujang, rice vinegar, sesame oil, minced garlic, grated ginger, and toasted sesame seeds until smooth. Toss the chicken thighs in the marinade, ensuring they are well-coated. Cover and let them marinate in the refrigerator for at least 20 minutes, up to 2 hours for a richer flavor.

In a spacious bowl, combine shredded green and red cabbage with julienned carrot for a fresh crunch. In a separate smaller bowl, whisk together mayonnaise, rice vinegar, sugar, salt, and pepper to create the dressing. Pour this dressing over the cabbage mixture, tossing until everything is evenly coated. Chill the slaw in the refrigerator while you cook the chicken.

Heat a cast-iron skillet or grill pan over medium-high heat until hot. Once hot, add the marinated chicken thighs, cooking for about 6-7 minutes on each side until caramelized and fully cooked through, reaching an internal temperature of 165°F. Once cooked, remove the chicken from the heat and let it rest for 5 minutes before slicing.

While the chicken is resting, prepare your buns. Spread butter generously on the cut sides of brioche or potato buns. Place them cut-side down in a skillet over medium heat and toast for 2-3 minutes, or until they are golden brown and slightly crispy, enhancing the sandwich's flavor and texture.

Begin to build your Korean BBQ Chicken Sandwich by layering sliced chicken thighs onto the toasted buns. For an extra kick, drizzle with additional BBQ sauce if desired. Top with a generous helping of the refrigerated cabbage slaw, placing the other half of the bun on top, and serve each sandwich warm for a delightful dining experience.