In a saucepan, whisk together the tapioca, milk, and salt. Bring to a simmer over medium-high heat, stirring frequently.

Reduce heat to low. Gradually whisk in the sugar. Cook uncovered for 20-30 minutes, stirring often, until tapioca pearls are plumped and mixture has thickened to a pudding-like consistency.

In a bowl, beat the eggs. Temper by slowly whisking in 1/2 cup of the hot tapioca mixture into the eggs.

Whisk the tempered eggs into the saucepan with the tapioca pudding. Increase heat to medium and cook for 2-3 minutes, stirring constantly, until thickened to a custard-like pudding texture. Do not boil.

Remove from heat and stir in the vanilla. Allow to cool slightly before serving warm, or chill thoroughly for a thicker set pudding.