Combine soy sauce, fish sauce, brown sugar, oyster sauce, shaoxing wine and lime juice in a small bowl.
Heat 2 teaspoons oil in a wok or large skillet over high heat. Add chicken; cook, stirring, until just cooked through, 5 to 7 minutes. Transfer to a plate.
Add the remaining 2 teaspoons oil, garlic, ginger and chili flakes to the pan. Cook, stirring, until fragrant, about 15 seconds. Toss in the red peppers and onions; cook, stirring for 6 minutes. Add mangoes and green onions; cook, stirring, for 1 minute. Add the reserved sauce and chicken; cook, stirring, until the sauce is thickened and the chicken is heated through, about 1 minute. Serve with rice, if desired.