Very thinly slice the chicken, then cut the slices into 1/2-inch pieces. Place the chicken in a shallow bowl, season with salt and pepper and sprinkle with the cornstarch. Toss and set aside.
Heat a very large skillet or a wok over medium high heat. Add half of the oil, then add the broccoli and sugar snap peas and sauté for two minutes. Add the peppers and stir fry for another 2 minutes. Turn the vegetable mixture into the serving platter or bowl you will be serving the stir fry in.
Return the pan to medium high heat and add the remaining half of the oil. Stir in the ginger and garlic, and then the chicken and stir fry for about 2 minutes until the chicken is mostly white, and almost cooked through. Return the vegetables to the pan along with the broth and bring to a simmer. Stir in the oyster and soy sauces along with the sugar, and simmer until the chicken is cooked through, the vegetables are crisp-tender, and the sauce is slightly thickened, about 2 minutes more.