In a small bowl, whisk together the soy sauce, hoisin sauce, chicken broth, cornstarch, garlic and ginger. Set the sauce aside.
Add 1 tablespoon oil to a large nonstick skillet set over medium-high heat. Add the broccoli and peppers and cook, stirring constantly, until crisp-tender, 3 to 4 minutes. Transfer the vegetables to a plate.
Add the remaining 2 tablespoons vegetable oil to the pan then add the chicken and cook, stirring frequently, until it is no longer pink.
Add the sauce and bring the mixture to a boil. Cook, stirring occasionally, until the sauce is thick and syrupy, about 2 minutes. Return the vegetables to the pan and toss to combine.
Serve the stir-fry with rice or noodles, or tuck it into lettuce wraps. Top with lime juice and serve.
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