In a small bowl, mix the soy sauce, oyster sauce, cornstarch, chicken broth, sesame oil, and sugar. Set aside.
Heat 1 tablespoon of vegetable oil in a large skillet or wok over medium-high heat. Add the sliced chicken and season with salt and pepper. 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 remaining tablespoon of vegetable oil. Add the minced garlic and ginger, and sauté for about 30 seconds until fragrant.
Add the broccoli florets to the skillet and stir-fry for 3-4 minutes until they are bright green and tender-crisp.
Return the cooked chicken to the skillet and pour the sauce mixture over everything. Stir well to combine and cook for an additional 2-3 minutes until the sauce thickens.
Serve hot over cooked rice or noodles.