Take a medium sized saucepan, and mix together your milk, tapioca, sugar, and salt. Stir.

Take this mixture to a boil over a medium heat and be sure to stir occasionally. Reduce the heat to low, cook and stir for 5 more minutes.

Take 1 cup of the hot milk mixture and whisk into the beaten eggs, 2 tablespoons at a time. Stir the egg mixture back into the tapioca now, until it is well mixed. Bring the pudding to a simmer over medium low heat and stir for 2 minutes longer, or until the pudding is thick enough to evenly coat a metal spoon. Remove the pudding from the heat, and now you can stir in the vanilla. Serve hot, or pour into serving dishes and refrigerate for later.