In a small bowl, whisk together soy sauce, oyster sauce, cornstarch, and water until smooth. Set aside.
Heat 1 tablespoon of vegetable oil in a large skillet or wok over medium-high heat.
Add the sliced chicken breast to the skillet and season with black pepper. Stir-fry for 4-5 minutes until the chicken is cooked through and lightly browned. Remove chicken from skillet and set aside.
Add the remaining 1 tablespoon of vegetable oil to the skillet.
Add the minced garlic and ginger to the skillet and stir-fry for 30 seconds until fragrant.
Add the sliced red and yellow bell peppers, julienned carrot, and broccoli florets to the skillet. Stir-fry for 3-4 minutes until the vegetables are tender-crisp.
Return the cooked chicken to the skillet with the vegetables.
Give the sauce mixture a quick stir and pour it over the chicken and vegetables.
Stir everything together and cook for 1-2 minutes until the sauce has thickened and evenly coats the chicken and vegetables.
Remove the skillet from heat and drizzle the sesame oil over the stir fry.
Garnish with sliced green onions and serve immediately with steamed rice or noodles.