Heat a large skillet over medium-high heat and add 1 Tbsp olive oil, swirling to coat the pan.
Add 1 lb lean ground beef and break it up with a spatula. Cook, stirring and breaking into pieces, until browned and no longer pink, about 5 minutes.
Season the beef with 1/2 tsp fine sea salt and 1/4 tsp pepper; stir to combine.
Add 1/2 onion (finely diced) and 1 carrot (julienned or coarsely grated). 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 to combine and continue to sauté, stirring occasionally, until the cabbage is tender and heated through, about 5–7 minutes. If the pan starts to stick, reduce heat to medium.
Taste and adjust seasoning with additional salt or pepper if desired.
Serve immediately, garnished with 1 Tbsp chopped green onion and 1/4 tsp sesame seeds, if using.