In a small bowl, soak 1 cup small tapioca pearls in 1 ½ cups of milk. Stir and place in the refrigerator for 3 hours.
After letting the tapioca soak, pour it into a medium saucepan. Add to the tapioca mixture 2 cups of milk, ⅔ cup sugar, salt, and egg yolks. Heat on medium heat, stirring frequently to keep milk from scalding. Once the tapioca mixture is simmering (do not boil), turn down the heat and simmer for 10 to 15 minutes, stirring constantly to keep the milk from scalding.
Carefully taste-test tapioca to be sure pearls are soft and no longer chewy. If you need to cook it longer, cook for an additional 5 minutes and check again; repeat until done. If your tapioca takes longer than 20 minutes, make sure you haven't turned the temperature down too low.
When the tapioca is done, remove it from the heat and let it cool for 15 minutes.
While the tapioca is cooling, beat egg whites with 1 tablespoon sugar until stiff peaks form.
Stir 1 teaspoon vanilla into the tapioca and gently fold in egg whites until completely incorporated.
Serve warm or chilled for 4 hours in the refrigerator. To serve, divide tapioca into individual bowls and top with fresh fruit of your choice.