Preheat oven to 350 degrees F (175 degrees C). Bring a large pot of water to a boil. Add kielbasa and boil for 8 to 10 minutes to reduce the amount of fat; drain. Prepare mashed potatoes according to package instructions.
Cut kielbasa into 4 equal portions, then split each lengthwise so that they lay flat but are still attached. Layer each sausage with sauerkraut and mashed potatoes. Top with grated cheese.
Bake in preheated oven for 10 to 12 minutes, or until heated through and cheese is melted.