In a small bowl, combine the chicken broth, soy sauce, brown sugar, cornstarch, sesame oil, ginger, garlic, and black pepper. Whisk until the sugar and cornstarch are dissolved. Set aside.
Season the chicken pieces with black pepper. Heat 1 Tbsp of oil in a large skillet or wok over medium-high heat. Add the chicken in a single layer and cook for 1 minute without stirring to get a good sear. Continue to cook, stirring, for 5 more minutes until golden and cooked through. Transfer to a bowl and cover to keep warm.
In the same skillet, add the remaining 1 Tbsp of oil, broccoli florets, onion, and mushrooms. Stir fry for 3 minutes until the vegetables are tender but still crisp.
Stir the sauce again to recombine, then pour it over the vegetables in the skillet. Simmer for 3-4 minutes until the sauce thickens. If the sauce is too thick, add water a tablespoon at a time to reach desired consistency.
Return the chicken to the skillet and stir to combine. Cook for an additional 30 seconds or until everything is heated through. Adjust soy sauce to taste, if necessary. Serve hot over steamed rice.