Heat 1 tablespoon oil in a 12-inch skillet over medium heat. Add chicken and cook, breaking it up with a wooden spoon, until no longer pink, about 5 minutes. Remove chicken from the skillet with a slotted spoon; set aside.
Pour remaining 1 tablespoon oil to the skillet. Add onion, carrots, and bell pepper; cook and stir until softened, about 4 minutes. Add ginger and garlic and cook for 2 minutes more.
Add ½ of the cabbage to the skillet. Cover and cook, stirring occasionally, until it begins to wilt, about 4 minutes. Add remaining cabbage. Cover and cook, stirring occasionally, until all the cabbage is tender and wilted, 6 to 8 minutes.
Whisk hoisin sauce, soy sauce, rice vinegar, honey, salt, and pepper together in a small bowl. Add sauce mixture and cooked chicken to the skillet. Cook and stir until heated through, about 2 minutes.
Garnish with sliced green onion before serving.
