Set a rack about 6 inches from the broiler and heat the broiler to high. Set out a 12-inch oven-safe skillet and a lid or sheet of foil for covering it.

Put the skillet on the stove over medium-high heat. Add the olive oil and heat it until it looks loose and shiny, about 1 minute.

Put the sausage links into the hot skillet. Cook them for 2 to 3 minutes per side until they are browned in spots, about 8 minutes total.

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

Lower the heat to medium. Add the sliced onion and 1/2 teaspoon of the salt to the skillet.

Cook the onion, stirring every minute or so, until it is soft and deep golden at the edges, 6 to 8 minutes. It should look glossy and a little sticky, not burnt.

Add the sliced potatoes and the black pepper to the skillet. Stir until the potato slices are coated with the onions and fat.

Pour 1/2 cup of the chicken broth into the skillet. Cover the skillet and cook for 8 minutes, until the potatoes have started to soften.

Uncover the skillet. Put the sliced cabbage into the skillet along with about three-quarters of the thyme leaves and the remaining 1/2 teaspoon salt.

Pour the remaining 1/2 cup broth around the cabbage. Cover the skillet again and cook for 6 to 8 minutes, until the cabbage has wilted down and looks tender.

Slice the browned sausages on the plate into 1-inch pieces. If any juices collected on the plate, keep them.

Return the sausage pieces and any juices from the plate to the skillet. Stir gently so the sausage is mixed through the vegetables without breaking up the potatoes.