In a small bowl, vigorously whisk the egg yolks with ¼ cup of the erythritol until pale and slightly thickened.
Set aside.
Attach a candy thermometer to the side of a large, heavy-bottomed pot.
Pour in the heavy cream and water, then stir in the remaining ¼ cup of erythritol.
Heat over medium, stirring occasionally, until the mixture is hot and the thermometer reads about 160°F (71°C).
Do not let it boil.
To temper the eggs, ladle about ½ cup of the hot cream mixture into the egg yolk bowl while whisking constantly. This gently raises the temperature of the eggs without scrambling them. Slowly pour the tempered egg mixture back into the pot, whisking the entire time to combine.
Continue cooking over medium heat, stirring frequently and scraping the bottom of the pot, until the custard thickens and the thermometer reaches 180°F (82°C), which should take about 20 minutes. The custard is ready when it coats the back of a spoon.
Immediately remove the pot from the heat.
Pour the custard through a fine-mesh strainer into a pitcher to catch any small bits of cooked egg.
Stir in the vanilla. For rapid chilling, place the pitcher in an ice water bath. Otherwise, let it cool before refrigerating until thoroughly chilled.
Serve cold in 8-ounce mugs or glasses. Store leftovers in an airtight container in the refrigerator for up to one week.
Don't skip the tempering step!
Pouring the hot liquid into the eggs slowly and whisking constantly is crucial to prevent the eggs from scrambling.