In a bowl, combine 2 tablespoons olive oil, soy sauce, minced garlic, grated ginger, and salt and pepper. Add chicken breasts, ensuring they are well coated. Cover and let marinate for at least 30 minutes, or up to 2 hours in the refrigerator.
While the chicken is marinating, steam the broccoli florets until they are bright green and tender-crisp, about 4-5 minutes. Set aside.
Preheat the grill or a grill pan over medium-high heat. Remove chicken from marinade, letting any excess drip off. Grill for about 6-7 minutes on each side, or until fully cooked through (internal temperature should reach 165°F). Allow the chicken to rest for 5 minutes before slicing.
In a small skillet, heat the remaining tablespoon of olive oil over medium heat. Add the minced garlic and red pepper flakes (if using) and sauté for about 1-2 minutes, until fragrant. Remove from heat and stir in the sesame oil.
In a large bowl, layer the cooked brown rice, steamed broccoli, and sliced grilled chicken. Drizzle the garlic sauce over the top.
Top the bowls with sliced green onions and a sprinkle of sesame seeds.