In a mixing bowl, combine the chicken pieces with soy sauce, oyster sauce, sesame oil, cornstarch, and a pinch of salt. Leave to marinate for about 10 minutes.
Heat the vegetable oil in a wok or large frying pan over medium-high heat.
Add the marinated chicken to the pan and stir-fry for 5-7 minutes or until the chicken is cooked through and lightly browned. Remove the chicken from the pan and set it aside.
In the same pan, add the sliced onion, bell pepper, garlic, and ginger. Stir-fry for about 3-4 minutes until the vegetables are tender.
Add the broccoli florets to the pan and stir-fry for another 3 minutes until they are bright green and crisp-tender.
Return the cooked chicken back to the pan, and sprinkle in the ground black pepper. Toss everything together for another minute to combine the flavors.
Taste and adjust seasoning with salt if necessary. Remove from heat.
Serve hot, garnished with chopped green onions.