Thinly slice the cabbage and prepare the carrot, green onions, garlic, and ginger.
In a small bowl, whisk together the soy sauce, sesame oil, rice vinegar, honey or brown sugar, and red pepper flakes if using.
Heat the cooking oil in a large skillet or wok over medium-high heat.
Add the garlic and ginger and stir for about 30 seconds, until fragrant.
Add the sliced cabbage and carrot to the skillet.
Stir fry for 5 to 7 minutes, tossing often, until the cabbage is tender but still slightly crisp.
Pour the sauce over the vegetables and toss well to coat.
Cook for another 1 to 2 minutes, until the sauce lightly coats the cabbage.
Stir in the sliced green onions.
Remove from heat and garnish with sesame seeds if desired.
Serve hot.