Set a 12-inch skillet with a lid over medium heat. Add the olive oil and let it warm for about 30 seconds.

Put the bratwurst links into the skillet. Cook for 2 to 3 minutes per side, until they are browned in spots.

Pour 1/4 cup of the chicken broth into the skillet. Cover the skillet and cook for 6 minutes, until the sausages feel firm and are almost cooked through.

Transfer the bratwurst from the skillet to a plate. Leave any fat in the skillet.

Put the potatoes into the same skillet. Sprinkle them with 1/2 teaspoon of the salt and 1/4 teaspoon of the black pepper.

Pour the remaining 1/2 cup chicken broth into the skillet. Cover and cook for 8 to 10 minutes, stirring once, until the potatoes are nearly tender when pierced with a fork.

Remove the lid from the skillet. Cook the potatoes for 4 to 5 minutes more, turning them every minute or so, until several sides are golden.

Add the onion and sliced fennel to the skillet. Cook for 4 minutes, stirring often, until the onion starts to turn clear and the fennel softens.

Add the cabbage to the skillet in a few handfuls. Sprinkle the cabbage with the remaining 1/2 teaspoon salt.

Stir the cabbage into the potatoes, onion, and fennel. Cook for 2 minutes, until the cabbage starts to wilt and shrink.

Cover the skillet again. Cook for 4 to 5 minutes, until the cabbage is tender but still has a little bite.

Move the bratwurst to a cutting board. Slice the sausages into 1-inch pieces.