Brown the sausages: Set a large lidded 12-inch skillet over medium heat. Add the olive oil and let it heat for about 30 seconds, until it looks loose and shiny.

Put the sausage links into the skillet. Cook them for 6 to 8 minutes, turning every few minutes, until they are browned on most sides but not fully cooked through.

Transfer the sausages to a plate. Leave the browned bits and any oil in the skillet.

Steam the potatoes: Add the potatoes to the skillet in an even layer. Sprinkle them with 1/2 teaspoon of the salt and 1/4 teaspoon of the pepper.

Pour 1 cup of the chicken broth into the skillet. Cover with the lid and cook over medium heat for 10 to 12 minutes, until the potatoes are almost tender when pierced with a knife.

Remove the lid. If more than a few tablespoons of broth are still in the skillet, cook uncovered for 1 to 2 minutes so most of the liquid cooks away.

Build the vegetables: Add the butter and sliced onion to the skillet with the potatoes. Cook for 4 to 5 minutes, stirring often, until the onion softens and picks up some brown edges.

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

Stir the cabbage into the onion and potatoes. Cook for 4 to 5 minutes, stirring often, until the cabbage is wilted and tender but still has a little bite.

Make the onion gravy: Use the back of a wooden spoon or a potato masher to lightly crush about half of the potatoes right in the skillet. Leave the rest whole so the finished dish has a mix of textures.

Sprinkle the flour over the vegetables in the skillet. Stir for 1 minute, until you no longer see dry flour.

Pour the remaining 1/2 cup chicken broth and the Worcestershire sauce into the skillet. Stir and scrape the bottom of the skillet until a light gravy forms, 1 to 2 minutes.