Fill a medium pot with water, add 1 teaspoon of the salt, and set it over high heat. Bring the water to a boil for the potatoes.

Set a large deep skillet, about 12 inches wide, over medium heat. Add the olive oil.

Put the sausage links into the skillet in a single layer. Cook for 4 to 5 minutes, until the first side is well browned.

Turn the sausage links with tongs. Cook for 3 to 4 minutes more, until the second side is browned. Transfer the sausages to a plate.

Add the sliced onion to the same skillet. Sprinkle in 1/2 teaspoon of the salt and cook for 3 minutes, stirring now and then, until the onion starts to soften.

When the water in the pot is boiling, add the potatoes. Boil them for 10 to 12 minutes, until a knife slides in with only a little resistance.

Add the sliced garlic and caraway seeds to the onion in the skillet. Stir for 30 seconds, just until fragrant.

Put the cabbage wedges into the skillet, cut-side down as much as possible. Cook for 3 to 4 minutes, until the cut sides pick up some browned spots.

Turn the cabbage wedges carefully. Cook the second cut side for 2 minutes.

Pour the chicken broth and apple cider vinegar into the skillet. Add the thyme sprigs and 1/4 teaspoon of the black pepper.

Return the browned sausage links to the skillet, nestling them between the cabbage wedges.

Cover the skillet and lower the heat to medium-low. Cook for 12 to 15 minutes, until the cabbage is tender and the sausage is cooked through; if you cut into the thickest link, the center should no longer look pink.