The rice should just barely bubble as it cooks. The milk may evaporate during cooking, so you may have to stir in a little hot water to keep the sauce-like consistency.
Cool the pudding in a jiffy by placing the saucepan in a bowl of ice water. Stir frequently so the pudding cools evenly.
Up to 24 hours ahead, make the pudding and spoon it into individual serving dishes; cover and refrigerate. Refrigerate the sauce in a covered container.