Brown the sausage: Heat the olive oil in a large pot over medium heat. Add the sliced sausage and cook until browned on both sides, about 4–5 minutes.
Transfer the sausage to a plate, leaving the drippings in the pot.
Sauté the aromatics: Add the onion, carrots, and celery to the pot. Season with a pinch of salt. Cook, stirring often, until softened, about 5–6 minutes.
Add the garlic and cook for 30 seconds until fragrant.
Add cabbage: Stir in the sliced cabbage. It will look like a lot, but it wilts down quickly. Cook for 3–4 minutes, tossing, until it begins to soften.
Build the broth: Return the sausage to the pot.
Add the chicken broth, diced tomatoes, thyme, smoked paprika, bay leaf, and a few grinds of black pepper. Stir well.
Simmer: Bring to a gentle boil, then reduce to a simmer. Cook uncovered for 15–20 minutes, or until the cabbage is tender and the flavors have melded.
Finish and taste: Stir in the apple cider vinegar.
Taste and adjust salt and pepper. Remove the bay leaf. Ladle into bowls and top with chopped parsley if you like.