Heat a large pot over medium-high heat. Add the sliced kielbasa and cook until lightly browned on both sides, 3–4 minutes per side. Remove and set aside, leaving rendered fat in the pot.

Reduce heat to medium. Add butter, then onion, celery, and carrots. Cook until onion is translucent, about 5 minutes. Add garlic and cook 30 seconds.

Return kielbasa to the pot with potatoes, chicken broth, bay leaf, and thyme. Bring to a simmer, then reduce heat and cook until potatoes are tender, about 12–15 minutes.

Remove bay leaf. Mash a cup of potatoes against the pot for thickness if desired. Stir in milk or cream and warm through. Season with salt and pepper to taste.

Ladle into bowls and top with shredded cheddar and chopped chives. Add black pepper and a drizzle of olive oil or butter if desired.