Make the sauce: In a small bowl, combine 3 tablespoons soy sauce or tamari, sugar, garlic oil, vinegar, sesame oil, and ginger. Set aside.
Combine ground pork or turkey with 1 tablespoon soy sauce and ¼ teaspoon pepper. Brown and crumble over medium-high heat. Drain fat and remove from skillet.
Add canola oil to the same skillet and heat over medium-high heat. Add carrots and stir-fry until crisp-tender, about 2 minutes, and then add the cabbage and stir-fry until it just starts to wilt, about 2 more minutes.
Add the browned meat and sauce mixture to the cabbage and carrots. Season with remaining ¼ teaspoon pepper and salt to taste. Stir and cook until most of the liquid is absorbed. Garnish with green onions and serve over rice if desired.