In a 10-inch skillet with a tight fitting lid, heat the olive oil over medium high heat. When the oil starts to glisten and moire, carefully add the sausages and cook, turning occasionally until browned. Transfer the sausages to a plate. Into the pan, over a medium heat, layer in half of the onion, potatoes and cabbage. Layer the remaining onions, potatoes and cabbage. Sprinkle with carrots and add 1 teaspoon of the thyme. Pour 3/4 cup of broth over the vegetables, sprinkle with salt and pepper and cover tightly. Simmer for 10 minutes.
After the vegetables have cooked for 10 minutes, nestle the sausages into the potato mixture, along with any accumulated juices. Add the remaining broth and thyme, cover and cook for an additional 10 minutes, or until potatoes and carrots are very tender. Remove the sausages and cut them into chunks. Return the sausages to the pan and serve.