Heat 1 teaspoon of oil in a large frying pan over medium heat. Add the broccoli and mushrooms and cook for approximately 4 minutes or until vegetables are tender.
Add the ginger and garlic to the pan and cook for 30 seconds more.
Remove the vegetables from the pan; place them on a plate and cover.
Wipe the pan clean with a paper towel and turn the heat to high. Add the remaining tablespoon of oil.
Season the chicken pieces with salt and pepper and add them to the pan in a single layer - you may need to do this step in batches. Cook for 3-4 minutes on each side until golden brown and cooked through.
Add the vegetables back to the pan and cook for 2 more minutes or until the vegetables are warmed through.
In a bowl whisk together the oyster sauce, chicken broth, sugar, sesame oil and soy sauce. In a small bowl mix the cornstarch with a tablespoon of cold water.
Pour the oyster sauce mixture over the chicken and vegetables; cook for 30 seconds. Add the cornstarch and bring to a boil; cook for 1 more minute or until sauce has just started to thicken.
Serve immediately, with rice if desired.