In a 10-inch skillet with a tight-fitting lid, heat the Olive Oil (2 tablespoon) over medium-high heat. When the oil starts to glisten and shimmer, carefully add the Irish Banger Sausage (1 pound) and cook, turning occasionally until browned.
Transfer the sausages to a plate. Into the pan over medium heat, layer in half of the Onion (1), Red Potato (0.5 pound) and Green Cabbage (1/2 head). Layer the remaining onions, potatoes, and cabbage. Sprinkle with Carrot (3) and Fresh Thyme (1 teaspoon).
Pour Low-Sodium Chicken Broth (3/4 cup) over the vegetables, sprinkle with Kosher Salt (1/2 teaspoon) and Freshly Ground Black Pepper (1/4 teaspoon) 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 Low-Sodium Chicken Broth (3/4 cup) and Fresh Thyme (1 teaspoon), then 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.