Prepare the instant white rice according to package directions using 2 cups water and 2 cups rice; keep warm.
In a medium bowl, whisk together 2/3 cup soy sauce, 1/4 cup brown sugar and 1 tablespoon cornstarch until smooth.
Add 1 teaspoon minced ginger, 1 tablespoon minced garlic and 1/4 teaspoon red pepper flakes (if using) to the sauce and mix to combine.
Reserve one-third of the sauce in a small bowl for later, then add the thinly sliced chicken breasts to the remaining sauce and stir to coat; refrigerate the marinated chicken for 30 minutes if time allows.
Heat 1 tablespoon sesame oil in a large skillet or wok over medium-high heat. Add broccoli florets, sliced carrots and bell pepper matchsticks and cook, stirring occasionally, about 5 minutes until just tender; remove vegetables to a plate.
Add the remaining 1 tablespoon sesame oil to the same skillet. Add the marinated chicken (discard any excess marinade left with the chicken) and cook over medium-high heat until browned and nearly cooked through, about 5 minutes.
Return the vegetables to the skillet, pour in the reserved one-third of the sauce, bring to a boil, and cook 5–7 minutes more until the chicken is cooked through and the sauce has thickened.
Serve the stir-fry over the prepared rice.