Place chicken in a large bowl. In a small bowl, whisk soy sauce, ginger, cornstarch, 2 cloves garlic and broth. Pour soy sauce mixture over chicken and toss to coat. Set aside to marinate at room temperature for 10 minutes (or cover and refrigerate for up to 4 hours).
Warm oil in a large nonstick skillet over medium-high heat. Add onion and bell pepper; cook, stirring constantly, until tender, about 3 minutes. Add remaining 1 clove garlic and cook, stirring often, for 1 minute longer. Add broccoli and cook, stirring frequently, until slightly softened, 3 to 5 minutes.
Pour chicken and marinade into skillet and cook, stirring, until chicken is no longer pink and sauce is warmed through and thickened, 5 to 7 minutes. Serve immediately, with brown rice and garnished with chopped roasted peanuts, if desired.