In a medium bowl soak tapioca in water overnight (about 8 hours). Drain water.
In a double boiler heat milk until no longer cold. Stir in salt and tapioca. Continue heating until small bubbles appear at the sides of the pan. Cover, turn heat to low and cook for 1 hour. Make sure the milk does not simmer or boil.
In a medium mixing bowl wisk together sugar and egg yolks until light in color. While whisking add a little bit of the hot milk mixture. Then add the egg yolk mixture to the hot milk mixture, stirring constantly. Place the double boiler over medium heat and cook until tapioca mixture is very thick, about 15 minutes.
Gently fold in beaten egg whites into the hot tapioca mixture. Stir in vanilla. Serve warm or chilled.