Wash tapioca in water, and drain. Soak the washed tapioca in 1 cup of the milk for about 6 to 8 hours.
Put 1 cup of the milk in a pot. Heat, then add tapioca and milk mixture. Add sugar. Cook slowly until thick. Beat egg and add to above mixture after it has cooled slightly.
Add butter and a dash of vanilla and pour into a greased baking dish. Bake at 325°F. for 45 minutes or until golden brown on top.