Heat a wok or large frying pan over high heat until extremely hot, 3 to 4 minutes. Add 2 tbsp. oil and swirl wok to coat. Add chicken. Cook undisturbed until browned, about 2 minutes. Add 2 tbsp. soy sauce. Cook until liquid begins to caramelize around chicken, stirring frequently, about 2 minutes. Transfer chicken to a plate.
Add remaining 2 tbsp. oil to wok and swirl to coat. Add bell pepper and garlic. Cook, stirring often, until garlic is golden brown, about 2 minutes. Add broccoli, sambal oelek, and hoisin; toss to coat. Cook until broccoli is tender-crisp, about 5 minutes.
Whisk together remaining 3 tbsp. soy sauce, the broth, and cornstarch; pour into wok and bring to a boil. Cook, stirring constantly, until sauce is thickened, 1 minute.
Stir in chicken and any juices. Stir-fry until chicken is cooked through and broccoli is tender, about 1 minute.
Serve over rice, sprinkled with peanuts.
*Find sambal oelek, a Southeast Asian-style hot sauce, in the international aisle of well-stocked grocery stores.
Wine pairing: V. Sattui 2015 Dry Riesling (Anderson Valley; $24)