Brown the butter: Melt 2 tablespoons unsalted butter in a large skillet or Dutch oven over medium heat. Let it cook undisturbed for about 1 minute until it just starts to turn golden and smell nutty.
Cook the onion and garlic: Add 1 yellow onion, thinly sliced and cook, stirring occasionally, until softened and lightly golden — about 5 minutes. Add 2 garlic cloves, minced and cook 1 minute more until fragrant.
Add cabbage and spices: Add 1.5 pounds green cabbage, cored and thinly shredded, 1 teaspoons smoked paprika, and 1 teaspoons caraway seeds. Toss everything to coat in the butter and spices. Cook for 3–4 minutes, stirring occasionally, until the cabbage begins to wilt and the paprika smells toasty.
Braise: Stir in 3 tablespoons apple cider vinegar, 0.5 cups chicken or vegetable broth, 1 tablespoons whole grain or Dijon mustard, 1 teaspoons sugar, 0.8 teaspoons salt, and 0.3 teaspoons black pepper. Reduce heat to medium-low, cover, and braise until the cabbage is tender — about 15 minutes.
Finish and serve: Uncover and cook 2–3 more minutes to evaporate any remaining liquid. Taste and adjust salt or vinegar. Serve hot.