Melt the butter in a large pot over medium heat. Add the thinly sliced red cabbage and toss to coat with the butter. Sauté until slightly wilted, about 5 minutes.
Sprinkle sugar over the cabbage and toss to coat evenly. Add the balsamic vinegar to the pot. Bring to a simmer, then reduce the heat to medium low.
Cover and simmer until the cabbage is completely tender but not mushy. Stir often, about 30 to 45 minutes total.
Season to taste with salt and pepper.
