Heat a wok over high heat. Add 1 tablespoon peanut oil and swirl it around until it gets hot. Add the garlic and ginger and stir-fry for about 30 seconds. Add the chicken; toss and stir until chicken is done. Remove from wok and set aside.

Add a little more oil to wok if needed. Add vegetables to wok; cook on high heat, tossing and stirring, until vegetables are heated through and the broccoli turns bright green. Stir the chicken broth mixture one more time to distribute the cornstarch evenly. Pour it over the vegetables, stirring until sauce thickens. Return chicken to wok and toss to coat chicken with sauce. Reduce heat to low. Cover and simmer for 2-3 minutes. Serve immediately with rice.