Heat oven to 375°F. Lightly grease a 9x13-inch baking dish.
Boil the sliced potatoes in well-salted water until just tender, 8–10 minutes. Drain and set aside.
Warm the olive oil in a large skillet over medium-high heat. Brown the kielbasa 4–5 minutes; remove to a plate.
Melt butter in the same skillet. Sauté onion until softened, 3–4 minutes. Stir in garlic for 30 seconds.
Sprinkle in flour and cook, stirring, 1 minute to form a roux.
Whisk in broth and milk gradually. Simmer, stirring, until slightly thickened, 2–3 minutes.
Take the pan off heat. Stir in sour cream, paprika, parsley, 1 tsp salt, pepper, and 1.5 cups cheddar until smooth.
Fold potatoes and browned kielbasa into the sauce until coated.
Transfer to the baking dish and scatter the remaining cheddar on top.
Bake until bubbling and lightly golden, 20–25 minutes. Broil 1–2 minutes for deeper color if desired.
Rest 10 minutes. Finish with green onions and the remaining salt to taste. Serve warm.