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

Add the olive oil to the skillet and swirl it to coat the bottom.

Put the bratwurst links into the skillet. Cook them for 6 to 8 minutes total, turning every few minutes, until they are browned on the outside. They do not need to be cooked through yet.

Transfer the browned bratwurst to a plate.

Put the potatoes and onion into the same skillet. Sprinkle them with 3/4 teaspoon of the salt.

Cook the potatoes and onion over medium heat for 5 minutes, stirring every minute or so, until the onion starts to soften and the potatoes pick up some brown spots.

Pour 1/2 cup water into the skillet. Cover with the lid and cook for 5 to 6 minutes, until the potatoes are starting to soften around the edges.

While the potatoes cook, cut the bratwurst on the plate into 1-inch pieces.

Uncover the skillet. Put the cabbage into the skillet a few handfuls at a time, stirring after each addition so it starts to wilt and make room in the pan.

Add the sliced apple to the skillet and stir it into the cabbage, onion, and potatoes.

Return the bratwurst pieces and any juices from the plate to the skillet.

Stir the whole-grain mustard with 1/4 cup water in a small cup until loosened. Pour that mustard mixture into the skillet.