In a small bowl, combine the soy sauce, oyster sauce, and hoisin sauce. Set aside.
Heat 1 tablespoon of vegetable oil in a large skillet or wok over medium-high heat.
Add the chicken slices and cook until browned and cooked through, about 5-6 minutes. Remove from the skillet and set aside.
In the same skillet, heat the remaining tablespoon of vegetable oil. Add the garlic and ginger, and cook until fragrant, about 1 minute.
Add the bell pepper, onion, broccoli, and carrot to the skillet. Cook, stirring constantly, until the vegetables are tender-crisp, about 3-4 minutes.
Return the cooked chicken to the skillet. Pour the sauce over the chicken and vegetables, and toss to coat everything evenly. Cook for an additional 1-2 minutes to heat through.
Season with salt and pepper to taste.
Serve the chicken stir fry hot over cooked rice or noodles. Garnish with chopped green onions and sesame seeds if desired.