In a bowl, combine soy sauce, brown sugar, sesame oil, rice vinegar, minced garlic, and grated ginger to create a marinade.
Add the chicken thighs to the marinade, ensuring they are well coated. Cover and let it marinate for at least 15 minutes (or up to 2 hours in the refrigerator for more flavor).
While the chicken is marinating, prepare the rice according to the package instructions if not using pre-cooked rice.
Preheat the grill or grill pan over medium heat. Remove the chicken from the marinade and place it on the grill. Cook for about 6-7 minutes per side, or until fully cooked and juices run clear. Discard any remaining marinade.
In the last few minutes of grilling, place broccoli florets on the grill. Cook until tender but still crisp, about 3-4 minutes.
Once the chicken is cooked, remove it from the grill and let it rest for a few minutes before slicing it into bite-sized pieces.
To assemble the bowls, start with a base of cooked rice. Top with sliced chicken, grilled broccoli, julienned carrots, and cucumber slices.
Finish by sprinkling chopped green onions and sesame seeds on top for garnish.