In a medium saucepan over medium heat, stir together the milk, sugar, and tapioca. Bring to a boil, stirring constantly. Once boiling, reduce the heat to low and continue to cook for another 5 minutes.

In a separate bowl, whisk 1 cup of the hot milk mixture into the beaten eggs, a few spoonfuls at a time until well combined, careful not to cook the eggs. Then stir the egg mixture back into the tapioca mixture until it is well incorporated.

Cook over low-medium heat until the mixture comes to a slow simmer and cook for another 2 minutes. Continue to stir often. The mixture should be thick and you should be able to coat the back of a spoon. Add the lemon zest and mix well. Remove from heat and add the vanilla extract.

Serve warm or pour into serving dishes and chill for several hours until cold.