Start the skillet: Set a large deep skillet with a lid over medium-high heat.

Brown the sausage: Add the olive oil to the skillet. When the oil looks shiny, add the sausage links and sprinkle them with about 1/4 teaspoon of the salt and a few grinds of the pepper.

Color the second side: Cook the sausage for 4 to 5 minutes, until the first side is browned. Turn the links and cook for 3 to 4 minutes more, until the second side is browned but the centers are not fully cooked yet.

Set the sausage aside: Transfer the sausage links to a plate. Leave the fat in the skillet.

Soften the onion: Lower the heat to medium. Add the sliced onion to the skillet and cook for 4 to 5 minutes, stirring now and then, until it softens and starts to pick up a little color.

Add the garlic and caraway: Stir in the sliced garlic and caraway seeds. Cook for about 30 seconds, just until fragrant.

Start the potatoes: Add the potato chunks, the remaining 1 teaspoon salt, and the rest of the pepper to the skillet. Stir to coat the potatoes in the onion mixture.

Pour in the braising liquid: Pour in the beer and chicken broth. Use a wooden spoon to scrape up any browned bits from the bottom of the skillet, then bring the liquid to a gentle simmer.

Cook until the potatoes start to soften: Cover the skillet with the lid and cook for 10 minutes, adjusting the heat as needed so the liquid stays at a steady simmer.

Add the sausage back: Uncover the skillet and nestle the sausage links into the potatoes. Pile the cabbage ribbons on top of everything in the skillet.

Wilt the cabbage: Cover the skillet again and cook for 8 to 10 minutes, until the cabbage is wilted and the sausage is cooked through. When you cut into the thickest link, the center should no longer be pink.

Finish the pan sauce: Transfer the sausage to a cutting board. Stir the cabbage and potatoes together in the skillet, then cook uncovered for 2 to 4 minutes, until the potatoes are fully tender and the liquid has reduced to a light sauce.