Cut the chicken breasts into bite-sized pieces. In a mixing bowl, combine the cornstarch, soy sauce, Sherry or Shaoxing wine, and oil. Add the chicken and mix until evenly coated. Let the chicken marinate for at least 10 minutes to tenderize.
While the chicken marinates, prepare the sauce by whisking together the low-sodium soy sauce, dark soy sauce, chicken broth, vinegar, Sherry or Shaoxing wine, hoisin sauce, sugar, and cornstarch until fully combined and smooth.
Heat a wok or large skillet over medium-high heat and add 2 tablespoons of oil. Once hot, add the marinated chicken and stir-fry until lightly golden and nearly cooked through, about 5–7 minutes. Remove the chicken from the pan and set aside.
Add the remaining 2 tablespoons of oil to the same pan. Add the dried chilies, bell peppers, ginger, and garlic. Stir-fry for about 4 minutes until fragrant and the peppers begin to soften.
Pour the prepared Kung Pao sauce into the pan with the vegetables and cook until the sauce thickens and coats the back of a spoon.
Return the chicken to the pan and toss everything together until well coated. Add the green onions, roasted peanuts, and sesame seeds if using. Stir once more and remove from heat. Serve hot.