Remove any damaged outer leaves from the cabbage, rinse thoroughly, cut into quarters, remove the core, and slice into thin strips.

Peel carrots and slice into thin ribbons. Peel and slice the onion.

Peel and finely mince the garlic.

Heat skillet over medium heat, add butter and oil, melt without browning.

Add sliced onions, cook for 4-5 minutes until softened and lightly golden.

Stir in carrot ribbons and cook for an additional 2-3 minutes.

Add minced garlic, cook for 30 seconds, then add half the cabbage.

Add remaining cabbage, toss to coat with oil, and sprinkle with seasonings.

Cook for 8-10 minutes, stirring until cabbage has lightly golden edges.

Pour in broth, cover, reduce heat and cook for 5 more minutes.

Remove lid, stir, increase heat to evaporate excess moisture.

Drizzle vinegar, add optional sugar, adjust seasonings, serve warm.