Whisk together the gochujang, soy sauce, brown sugar, sesame oil, garlic, ginger, rice vinegar, mirin, and red pepper flakes in a bowl.
Add the chicken pieces and toss to coat them evenly. Marinate for at least 30 minutes (or up to 4 hours refrigerated).
Make the spicy sauce by whisking together gochujang, mayonnaise, sesame oil, lime juice, garlic, and honey. Add water 1 tablespoon at a time until you reach a pourable consistency. Set aside.
Heat the vegetable oil in a large cast iron skillet over medium-high heat.
Working in batches, cook the chicken pieces 5-6 minutes per side until caramelized and cooked through, maintaining the heat for grill marks. Don't crowd the pan.
Chop the chicken into 1-inch slices.
To assemble, divide the cooked rice among four bowls. Arrange the chicken pieces on top. Add carrots, cucumber, and scallions in sections around the bowl. Drizzle with spicy sauce and top with sesame seeds.
Serve assembled Korean barbecue chicken rice bowls.