Heat the vegetable oil in a large skillet or wok over medium-high heat.
Season the chicken pieces with salt and black pepper. Add the chicken to the skillet 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 about 3-4 minutes until the vegetables are tender-crisp.
In a small bowl, mix the soy sauce, oyster sauce, cornstarch, and chicken broth until well combined.
Return the cooked chicken to the skillet and pour the sauce mixture over it. Stir well to coat the chicken and vegetables. Cook for an additional 2-3 minutes until the sauce thickens.
Stir in the chopped green onions just before serving.
Serve the black pepper chicken over cooked rice or noodles.