Brown the sausage: Set a large deep skillet with a lid over medium-high heat and let it heat for about 2 minutes.

Add 1 tablespoon of the olive oil to the skillet.

Add the sliced andouille sausage to the skillet in a single layer.

Cook the sausage for 4 to 5 minutes, stirring once or twice, until the edges are browned.

Transfer the sausage to a plate and leave the drippings in the skillet.

Start the vegetables: Add the remaining 1 tablespoon olive oil to the skillet.

Put the halved potatoes into the skillet cut-side down.

Cook the potatoes for 5 minutes without moving them much, until the cut sides have some golden color.

Add the sliced onion to the skillet and cook for 2 minutes, stirring so it starts to soften.

Add the cabbage strips to the skillet and cook for 3 minutes, stirring often, until the cabbage starts to wilt and make room in the pan.

Sprinkle in the 1 teaspoon kosher salt and 1/2 teaspoon black pepper.

Build the braise: Add the sliced garlic and the tomato paste to the skillet.