Preheat oven to 350 degrees F (175 degrees C). Butter a 1.5 quart baking dish.
In a medium bowl, combine the potatoes, butter, eggs, onion, salt and paprika. Mix well.
Place potato mixture into the prepared baking dish and pour milk over top. Bake in the preheated oven for 40 minutes. Sprinkle top with cheese, return to oven and bake until cheese melts and is slightly browned.