Slice the chicken breasts thinly across the grain. In a small bowl, combine the sliced chicken with 1 tablespoon of the soy sauce and set it aside to marinate briefly.
In another bowl, mix the remaining soy sauce with the oyster sauce (if using), sesame oil, rice vinegar, and the cornstarch slurry. Whisk until the sauce is smooth and well-blended.
Warm 1 tablespoon of the vegetable oil in a large skillet or wok over medium-high heat. Add the marinated chicken and stir-fry until it is just cooked through and lightly browned, about 4 to 5 minutes. Remove the chicken from the pan and set it aside.
Add the remaining tablespoon of vegetable oil to the pan. Stir in the minced garlic, grated ginger, and red pepper flakes. Cook briefly until aromatic, about 30 seconds.
Add the shredded cabbage, julienned carrot, and sliced bell pepper to the pan. Stir-fry for 3 to 4 minutes until the vegetables are bright and crisp-tender.
Return the cooked chicken to the pan. Pour the prepared sauce over the ingredients, tossing to coat everything evenly. Stir continuously for about 2 minutes, until the sauce slightly thickens and everything is well combined.
Sprinkle the chopped green onions and toasted sesame seeds over the top. Serve the stir-fry hot, accompanied by steamed rice or your preferred noodles.