Grill the kielbasa on both sides until an internal temperature of 160 degrees is reached. Remove from the heat and set aside.
Heat the oil in a non-stick pan or skillet over medium heat. Add the potatoes and cook for 5 to 7 minutes until they start to brown. Add the cabbage, onions, and garlic. Cook while stirring frequently, for about 5 minutes until the cabbage starts to wilt. Add the vinegar and broth and simmer until most of the liquid evaporates.
Season with salt and pepper. Add in the butter and stir.
Serve the Kielbasa over the cabbage and potatoes with crusty bread.