In a large bowl whisk the marinade ingredients. Transfer ½ cup of the marinade to small bowl and set aside to use as a glaze when the food comes off the grill. Add the chicken pieces to the remaining marinade in the large bowl and toss to coat well. Cover and refrigerate for 1 to 2 hours.
If using bamboo skewers, soak in water for at least 30 minutes.
Prepare the grill for direct cooking over medium heat (350° to 450°F) and preheat the grill pan.
Thread the chicken pieces onto skewers, leaving space in between each piece. Brush the chicken with any leftover (not reserved) chicken marinade from the large bowl.
Toss the broccoli and mushrooms with enough oil to lightly coat them and season evenly with salt and pepper.
Brush the cooking grates clean. Grill the chicken over direct medium heat, with the lid closed as much as possible, until it is firm to the touch and opaque all the way to the center, 8 to 12 minutes, turning once or twice. At the same time, spread the broccoli and mushrooms in a single layer on the grill pan and cook over direct medium heat until crisp-tender, about 6 minutes, turning occasionally. Remove from the grill as they are done. Immediately brush the chicken and vegetables with the reserved ½ cup glaze. Top with the sesame seeds and serve warm with steamed brown rice, if desired.