Add pistachios to the bowl of a food processor. Process until the pistachios are finely ground. Add sugar and water and continue to process to form a paste.
Add pistachio paste and milk to a small saucepan. Heat over medium heat until steaming, stirring often to prevent scorching.
Meanwhile, whisk together egg yolks, remaining ¼ cup sugar, cornstarch and salt in a medium bowl or 2 cup measuring cup (it will be thick).
Stir ½ cup of the hot milk mixture into the egg mixture. Add another ½ cup of the milk mixture and stir until smooth.
Transfer the egg mixture into the remaining milk mixture, and continue to heat over medium until the mixture comes to a boil. Continue to simmer until thickened, about 2-3 more minutes, stirring constantly. Remove from heat and stir in vanilla.
Transfer pudding to 6 individual dessert dishes, or one larger serving bowl. Cover with plastic wrap, so that the plastic is actually touching the pudding. Chill completely, at least 2 hours.