Halve cabbage and cut out core. Thinly slice cabbage halves into ¼ to ½ inch slices; cut across slices for bite-size pieces. You should have about 8 to 10 cups.
In a large deep skillet or Dutch oven, over medium heat, melt butter and olive oil together. When butter is melted, add cabbage and season with salt and pepper. Stir well. Cook until wilted, about 5 minutes, stirring frequently.
Add vinegar and sugar. Bring to a simmer, reduce heat to low. Cover, and cook for 30 minutes or until tender, stirring frequently. If cabbage starts to stick to bottom of the pan, add water, a tablespoon at a time.
Taste and season with more salt and pepper, if needed. Serve immediately.
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Notes
Veganand dairy-free: Use 2 tablespoons olive oil; omit the butter.
Flavor variations: Add an onion or apple, thinly sliced, to cook with the red cabbage. Stir in a half teaspoon of caraway seeds, if desired. Use balsamic vinegar instead of red vinegar.
To freeze: Cooked red cabbage can be frozen in freezer safe bags for up to 3 months. There's no need to thaw it before reheating.