Heat sesame oil in a large skillet or wok over medium-high heat.
Add ground beef, breaking it up with a spatula. Cook until browned, about 5–7 minutes. Drain excess fat if necessary.
Add garlic and ginger to the skillet, stir-fry for 1 minute until fragrant.
Toss in the cabbage and carrots. Stir-fry for 5–7 minutes, or until the vegetables are tender but still crisp.
Pour in soy sauce, rice vinegar, and hoisin sauce if using. Mix well to coat all ingredients.
Season with crushed red pepper flakes, salt, and pepper to taste.
Remove from heat. Sprinkle with green onions and sesame seeds before serving.
