Combine 3/4 cup water and tapioca in a medium saucepan; let stand 30 minutes.
Add milk and salt to tapioca mixture; bring to a boil. Reduce heat, and simmer 30 minutes or until mixture thickens, stirring frequently. Combine sugar and eggs, stirring well with a whisk. Gradually stir about 1/2 cup hot tapioca mixture into egg mixture. Return tapioca mixture to pan. Cook 4 minutes or until thickened, stirring constantly.
Remove from heat; stir in vanilla. Spoon pudding into a bowl. Place bowl in a large ice-filled bowl; let stand 15 minutes or until pudding is cool, stirring occasionally. Cover surface of pudding with plastic wrap. Chill. Garnish with grated nutmeg, if desired.