In a saucepan over medium heat, stir together the milk, tapioca pearls and sugar. Bring the mixture to a boil, stirring constantly so the milk does not scorch.
Once the mixture is boiling, reduce the heat to low. Cook and stir for an additional 5 minutes.
Crack the eggs into a bowl and use a fork to whisk them together.
Remove 1 cup of hot milk mixture. Add two tablespoons of the hot milk mixture into the beaten eggs. Whisk well. Continue to add the hot milk, 2 tablespoons at a time, whisking well with each addition.
Stir the egg mixture back into the saucepan with the hot milk. Cook and stir for 2 minutes over medium heat until the pudding thickens. When it is done it should coat the back of a metal spoon.
Remove the pudding from the heat and stir in the vanilla.