Heat the vegetable oil in a large skillet or wok over medium-high heat. Add the chicken pieces and cook for about 5-7 minutes, or until they are golden brown and cooked through. Remove the chicken from the skillet and set aside.
In the same skillet, add the minced garlic and ginger. Sauté for about 30 seconds until fragrant.
Add the shredded cabbage, bell pepper, and carrots to the skillet. Stir-fry for about 5-6 minutes, or until the vegetables are tender but still crisp.
Return the cooked chicken to the skillet. Pour in the soy sauce, rice vinegar, and sesame oil. Stir everything together and cook for an additional 2-3 minutes to heat through. Season with salt and pepper to taste.
Remove from heat and garnish with sliced green onions and sesame seeds if desired. Serve immediately.