Start the skillet: Set a large skillet with a lid over medium heat. Add the olive oil and let it warm for about 30 seconds.

Brown the sausage: Put the sausage links into the skillet in a single layer. Cook without moving them for 3 to 4 minutes, until the bottoms are browned.

Turn the sausage links with tongs. Cook for 3 to 4 minutes more, until the second side is browned, even though the centers will still be undercooked.

Transfer the browned sausage links to a plate. Keep the skillet over medium heat.

Cook the potatoes: Put the halved potatoes cut-side down into the skillet. Sprinkle them with 1/2 teaspoon of the salt.

Cook the potatoes for 5 minutes, until the cut sides have some golden brown spots. Turn them and cook 2 minutes more.

Add the vegetables: Add the sliced red onion to the skillet with the potatoes. Cook for 2 minutes, stirring once or twice, until the onion starts to soften.

Add the sliced red cabbage to the skillet a few handfuls at a time. Stir after each addition so the cabbage starts to wilt and fit in the pan.

Sprinkle the cabbage mixture with the remaining 1/2 teaspoon salt, the black pepper, the caraway seeds, and the brown sugar. Stir well so the seasonings coat the vegetables.

Build the braise: Pour the apple cider vinegar and chicken broth into the skillet. Use a spoon to loosen any browned bits stuck to the bottom of the pan.

Nestle the sausage links back into the skillet, placing them on top of and between the vegetables. Any juices on the plate can go into the skillet too.

Cover the skillet with the lid and lower the heat to medium-low. Cook for 12 to 15 minutes, until the potatoes are easy to pierce with a knife, the cabbage is soft, and the sausage is cooked through with no pink in the center.