Rinse tapioca pearls and then pour into a small bowl and cover with water. Let soak 45-60 minutes.
Add milk and first ¾ cup of sugar to a large saucepan. Bring mixture to a simmer over medium heat. Once small bubbles form, add the tapioca pearls and reduce heat to a gentle simmer. Stir frequently and cook for 15-20 more minutes or until pearls become translucent.
As pearls cook, beat together the eggs and remaining ¾ cup sugar in a medium size bowl.
Slowly pour a stream of the hot mixture into the eggs about a ⅓ cup at a time, stirring constantly. Continue until half the hot mixture has been beaten into the eggs.
Slowly pour the contents of the egg bowl back into the saucepan, continuously stirring.
Cook for an additional 3-5 minutes or until the pudding starts to thicken (it will continue to thicken as it cools as well).
Remove from heat and stir in vanilla. Transfer to serving dishes and chill until serving.