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

Put the sausage links into the skillet in a single layer. Cook them for 3 to 4 minutes, until the bottoms are browned.

Turn the sausage links with tongs. Cook for 3 more minutes, until the second side is browned.

Pour 1/4 cup of the chicken broth into the skillet. Cover with the lid and cook for 4 to 5 minutes, until the sausage is cooked through and no longer pink in the center when one link is cut open.

Transfer the sausage links from the skillet to a plate. Leave the fat and browned bits in the skillet.

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

Pour the remaining 1/2 cup chicken broth around the potatoes. Cover the skillet and cook for 8 minutes over medium heat so the potatoes soften.

Remove the lid. Cook the potatoes for 4 to 6 minutes, turning them once or twice, until the liquid has cooked away and the cut sides are golden in spots.

Add the mushrooms to the skillet. Cook for 4 minutes, stirring now and then, until they shrink and start to brown.

Add the sliced garlic to the skillet. Stir for 30 seconds, just until it smells fragrant.

Add the cabbage to the skillet. Sprinkle with the remaining 1/2 teaspoon kosher salt and the remaining 1/4 teaspoon black pepper.

Use tongs or a large spoon to toss the cabbage with the potatoes and mushrooms. Cover the skillet and cook for 4 minutes so the cabbage starts to soften.