Heat a large skillet over medium-high heat and add 1 Tbsp olive oil; heat until the oil shimmers.
Add 1 lb lean ground beef to the hot skillet. Break the meat up with a spatula and season with 1/2 tsp fine sea salt and 1/4 tsp pepper. Cook, stirring occasionally and breaking up any large pieces, until the beef is no longer pink, about 5 minutes.
Add 1/2 onion (finely diced) and 1 carrot (julienned or coarsely grated) to the beef. Sauté, stirring occasionally, until the onion is tender, about 5–7 minutes.
Add 3 garlic cloves (minced) and cook, stirring constantly, for about 30 seconds until fragrant.
Add 3 cups cabbage (thinly sliced), 1 tsp ground ginger, 1/4 cup low-sodium soy sauce, 2 tsp sesame oil, and 1/2 tsp granulated sugar. Stir well to combine.
Continue sautéing, stirring occasionally, until the cabbage is tender and most of the liquid has reduced, about 5–7 minutes. If the pan splatters when you add the soy sauce, reduce the heat to medium.
Taste and adjust seasoning with additional salt and pepper if desired.
Serve immediately, garnished with 1 Tbsp chopped green onion and 1/4 tsp sesame seeds, if using.