Start by peeling and dicing the potatoes into evenly sized cubes. Place them in a large pot of salted water and bring to a boil. Cook for about 5 to 7 minutes until slightly tender but not fully cooked, then drain and set aside.
Melt the butter in a large skillet over medium heat. Add the sliced kielbasa and cook until the edges become golden and slightly crispy. Remove the sausage and place it on a plate while keeping the flavorful drippings in the skillet.
Add the diced onion to the same skillet and cook until soft and translucent. Stir in the minced garlic and cook briefly until fragrant. Transfer the mixture to a large mixing bowl.
Add the cream of mushroom soup, sour cream, milk, Dijon mustard, Worcestershire sauce, salt, and black pepper to the bowl. Whisk everything together until the sauce becomes smooth and well combined.
Fold the parboiled potatoes, browned kielbasa, and 1½ cups of shredded cheddar cheese into the sauce. Stir gently until every piece is coated with the creamy mixture.
Grease a 9x13-inch baking dish with butter. Spread the potato and sausage mixture evenly into the dish. Sprinkle the remaining cheddar cheese across the top for a perfectly cheesy finish.
Place the casserole in a preheated 350°F (175°C) oven. Bake for 35 to 40 minutes until bubbly around the edges and golden brown on top.
Allow the casserole to rest for about 10 minutes before serving. This helps the sauce thicken slightly and makes serving much easier.