Heat the vegetable oil in a large skillet or wok over medium-high heat. Add the sliced chicken, season with salt and pepper, and cook until browned and cooked through, about 5-7 minutes. Remove the chicken from the skillet and set aside.
In the same skillet, add the green onions, garlic, and ginger. Sauté for about 1 minute until fragrant.
In a small bowl, whisk together the soy sauce, brown sugar, cornstarch, and water until smooth. Pour this mixture into the skillet and bring to a simmer.
Add the broccoli and bell peppers to the skillet. Stir to coat the vegetables in the sauce and cook for about 3-4 minutes until the vegetables are tender-crisp.
Return the cooked chicken to the skillet and toss everything together until heated through and well combined. If desired, sprinkle with red pepper flakes for added heat.
Serve hot over rice or noodles.