In a large bowl, combine the chicken pieces with salt, black pepper, soy sauce, oyster sauce, and cornstarch. Mix well and let marinate for at least 15 minutes.
Heat the vegetable oil in a large skillet or wok over medium-high heat. Add the marinated chicken 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 minced garlic and ginger. Sauté for about 30 seconds until fragrant.
Add the sliced bell pepper and broccoli florets to the skillet. Stir-fry for 3-4 minutes until the vegetables are tender-crisp.
Return the chicken to the skillet and pour in the chicken broth. Stir everything together and cook for an additional 2-3 minutes until the sauce thickens slightly.
Stir in the chopped green onions just before serving. Serve hot over cooked rice.