In a medium bowl, whisk together the soy sauce, brown sugar, honey, gochujang, rice vinegar, sesame oil, garlic, ginger, and sesame seeds until the sugar dissolves. Add the chicken thighs and toss to coat evenly. Let the chicken marinate at room temperature for at least 20 minutes, or for deeper flavor, cover and refrigerate for up to 2 hours.

While the chicken marinates, prepare the slaw. In a large bowl, combine the shredded green cabbage, red cabbage, and grated carrot. In a small bowl, whisk together the mayonnaise, rice vinegar, sugar, salt, and pepper. Pour the dressing over the cabbage mixture and toss well to coat. Cover and refrigerate until ready to serve; the cool crunch balances the spicy-sweet chicken.

Heat a cast-iron skillet or grill pan over medium-high heat. Lightly oil the pan if needed. Add the marinated chicken thighs in a single layer (work in batches if necessary) and cook for 6–7 minutes per side, until nicely charred and cooked through (internal temperature reaches 165°F). The sugars in the marinade will caramelize beautifully.

Transfer the cooked chicken to a cutting board and let rest for 5 minutes before slicing crosswise into strips or serving whole on the sandwich.

Split the buns in half and spread a thin layer of softened butter on the cut sides. Toast them in the same skillet over medium heat (or under the broiler) until golden and crisp on the edges, about 1–2 minutes. This step prevents sogginess and adds a rich layer of texture.

Place the bottom bun halves on plates. Layer on the sliced or whole grilled chicken, spoon over a little extra gochujang or Korean BBQ sauce if desired, then top with a generous handful of the chilled cabbage slaw. Crown with the top bun and serve immediately while hot.