Heat the oil in a large Dutch oven over medium-high heat. Add the sliced sausage and cook until browned on both sides, about 5 minutes. Don't rush this step—those caramelized edges are pure flavor gold.
Transfer the sausage to a plate using a slotted spoon, leaving the flavorful oil behind. Don't clean that pot! Those browned bits are your flavor foundation.
Add the onions, carrots, and celery to the pot. Cook until the onions begin to soften, about 5 minutes, stirring occasionally to scrape up those tasty browned bits from the bottom.
Toss in the garlic and cook for another 30 seconds until fragrant. Your kitchen should be smelling amazing right about now.
Add the cabbage, bay leaves, caraway seeds, thyme, and black pepper. Cook, stirring occasionally, until the cabbage begins to wilt down, about 5 minutes.
Pour in the beer, using it to deglaze the pot. Scrape the bottom with your wooden spoon to get all those flavorful bits incorporated into the liquid. Let it simmer for about 2 minutes to cook off some of the alcohol.
Add the chicken broth, diced tomatoes with their juice, and potatoes. Give everything a good stir to combine.
Return the browned sausage to the pot, along with any accumulated juices. Bring the stew to a boil, then reduce heat to maintain a gentle simmer.
Cover and cook for about 30-40 minutes, or until the potatoes and vegetables are tender.
Stir in the mustard, brown sugar, and apple cider vinegar. These ingredients add that perfect balance of tangy, sweet, and savory that makes German food so crave-worthy.
Taste and adjust seasonings, adding salt as needed. Remember that the sausage and broth already contain salt, so add gradually.
Remove the bay leaves before serving. Ladle the hot stew into bowls, garnish with fresh parsley, and offer sour cream on the side.