Brown the sausage. Heat a large pot or Dutch oven over medium heat. Add the sausage and cook until nicely browned on both sides, about 5–7 minutes.
Transfer to a plate, leaving any fat in the pot.
Sauté aromatics. Add butter to the pot. Stir in onion, leeks, carrots, and celery with a pinch of salt. Cook until softened and lightly golden, 6–8 minutes.
Add garlic and cook 30 seconds until fragrant.
Bloom spices. Sprinkle in caraway seeds and stir for 20–30 seconds. This wakes up their flavor without burning.
Add potatoes and broth. Stir in potatoes, then pour in the broth. Add the bay leaf.
Bring to a boil, then reduce to a simmer.
Simmer until tender. Cook uncovered for 15–20 minutes, stirring occasionally, until the potatoes are fork-tender.
Partially mash for body. Use a potato masher to gently mash a few scoops of potatoes in the pot. Don’t overdo it—just enough to thicken the soup naturally.
Stir in sausage and enrich. Return the sausage to the pot. Add mustard and cream. Simmer 3–5 minutes to warm through and meld flavors.
Adjust thickness with a splash of broth or cream as needed.
Brighten and season. Turn off the heat. Stir in vinegar and fresh dill or parsley. Taste and season with salt and pepper.
The vinegar should lift the flavor without making the soup tangy.
Serve. Ladle into bowls and top with chives if you like. Serve with crusty bread for dipping.