In a large non-reactive pot or Dutch oven over medium heat, sauté the sausage until beginning to brown, about 5 minutes. Add the cabbage, onions, and carrot. Cover and cook until the vegetables begin to wilt, stirring occasionally, about 10 minutes.
Stir in the sauerkraut and all the remaining ingredients except the salt. Reduce heat to low, cover and simmer for 1½ -2 hours, until everything is quite tender. Add more beer if needed to keep the vegetables moist during cooking.