Set a large skillet with a lid over medium heat and let it warm for about 1 minute.

Brown the sausage: Put the andouille rounds into the dry skillet in a single layer.

Cook the sausage for 3 to 4 minutes, until the bottoms are browned and some fat has come out into the skillet.

Turn the sausage rounds and cook for 2 to 3 minutes more, until the second side is browned.

Build the base: Transfer the sausage to a plate and leave any fat in the skillet.

Add the olive oil to the skillet if it looks dry.

Add the onion, bell pepper, and celery to the skillet. Cook for 3 minutes, stirring often, until the vegetables start to soften.

Add the garlic and Cajun seasoning to the skillet. Stir for 30 seconds, just until fragrant.

Cook the potatoes: Add the potatoes, the chicken broth, and 1/2 teaspoon of the salt to the skillet. Stir so the potatoes are coated with the seasoned liquid.

Cover the skillet with the lid and cook for 8 to 10 minutes over medium heat. Stir once halfway through. The potatoes should be almost tender when you poke one with a fork.

Smother the cabbage: Uncover the skillet and add the sliced cabbage and the remaining 1/2 teaspoon salt. Toss to start wilting the cabbage. If the skillet is very full, add the cabbage in two handfuls and let the first handful shrink before adding the rest.

Cover the skillet again and cook for 4 minutes, until the cabbage has softened and dropped in volume.