In a medium saucepan, combine milk, tapioca, sugar, and salt. Bring the mixture to a gentle boil over medium heat, stirring constantly.

Reduce heat to low and simmer for 5 minutes, continuing to stir, until the tapioca pearls are fully translucent and the mixture has thickened.

Gradually whisk in the beaten eggs to the saucepan, ensuring to stir quickly to incorporate without curdling the eggs.

Continue to cook over medium-low heat for another 2-3 minutes, stirring constantly, until the pudding is thick enough to coat the back of a spoon.

Remove from heat and stir in the vanilla extract. Let the pudding cool for a few minutes before serving warm, or chill in the refrigerator if preferred cold.