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In a large nonstick skillet, melt butter over medium-high heat. Add the sliced sausage and cook until browned, 4–5 minutes. Transfer to a plate and set aside.
Reduce the heat to medium and in the same skillet, add the onion. Cook until slightly softened, about 3 minutes. Stir in the garlic and cook for 30 seconds more.
Add the chopped cabbage, broth, carrot (if using), vinegar, salt, pepper, and smoked paprika. Cover and cook, stirring occasionally, until the cabbage begins to soften, about 15 minutes.
Uncover, add the sausage to the skillet, and toss to combine. Cook for another 5 to 10 minutes, or until the liquid has evaporated and the cabbage is tender.
Season to taste and serve.