Combine milk, sugar, tapioca and salt in a medium saucepan. Let stand for 10 minutes.
Bring milk mixture to a boil over medium heat, stirring constantly. Simmer for 2 minutes, stirring frequently.
Beat eggs in a medium bowl. Slowly stir half of the pudding mixture into eggs, whisking quickly to prevent eggs from scrambling, then stir the pudding mixture back into the saucepan.
Cook pudding, stirring, until it begins to thicken, about 2 minutes. Remove from heat. Add vanilla. Cool slightly in saucepan.
Divide among four bowls and serve immediately or chill.