Brown the beef: Heat the olive oil in a large skillet over medium-high heat. Add the ground beef and cook for about 5 minutes, breaking it up with a spatula until it is browned and cooked through. Season with salt and pepper.
Sauté aromatics: Stir in the diced onion and grated carrot. Cook for 5 to 7 minutes, stirring occasionally, until the onion softens.
Add garlic: Add the minced garlic and cook for just 30 seconds until fragrant, being careful not to burn it.
Wilt the cabbage: Add the sliced cabbage, ground ginger, soy sauce, sesame oil, and sugar. Sauté for another 5 to 7 minutes until the cabbage is tender but still has a slight crunch.
Serve: Remove the pan from heat. Top with chopped green onions and sesame seeds if desired, and serve warm.