Heat vegetable oil in a large skillet over medium-high heat.
Add sliced chicken breast and cook until golden brown, about 5-7 minutes.
Remove chicken from the skillet and set aside.
In the same skillet, add minced garlic and ginger, sauté for 1 minute.
Add broccoli florets and stir-fry for 3-4 minutes until bright green.
Return the chicken to the skillet with broccoli.
Pour soy sauce and oyster sauce over the chicken and broccoli.
Stir well to combine and cook for another 2-3 minutes.
Drizzle with sesame oil and season with salt and pepper to taste.
Serve hot over cooked rice or noodles.