Brown the sausage: Set a large 12-inch skillet with a lid over medium-high heat.

Add the Italian sausage to the hot skillet. Cook for 5 to 6 minutes, breaking it into bite-size pieces with a spoon, until it is browned and no large pink pieces remain.

Transfer the sausage to a plate with a slotted spoon. Leave the fat in the skillet.

Start the potatoes: If the skillet looks dry, add the olive oil. Add the potatoes and 1/2 teaspoon of the salt to the skillet.

Stir the potatoes so they are coated in the fat and oil.

Pour 1/2 cup of the chicken broth into the skillet. Cover with the lid, lower the heat to medium, and cook for 8 minutes, until the potatoes are starting to soften when pierced with a knife.

Remove the lid. Cook the potatoes for 4 to 5 minutes, stirring every minute or two, until the liquid is gone and the cut sides pick up some brown spots.

Soften the vegetables: Add the fennel and onion to the skillet. Cook over medium heat for 4 minutes, stirring often, until they soften and start to turn lightly golden at the edges.

Add the garlic and cook for 30 seconds, stirring, just until it smells fragrant.

Add the cabbage, the remaining 1/2 teaspoon salt, and the black pepper. Toss for 1 to 2 minutes, until the cabbage starts to wilt and fits more easily in the skillet.

Pour in the remaining 1/4 cup chicken broth. Cover the skillet and cook for 4 minutes, until the cabbage softens further.

Remove the lid and stir. Cook for 2 to 3 minutes more, until the cabbage is tender and most of the liquid has cooked away.