In a medium saucepan, heat the heavy cream and instant espresso powder over medium heat, stirring until the espresso dissolves and the cream just begins to simmer. Do not boil.
In a medium bowl, whisk the egg yolks, erythritol, and salt together until pale and slightly thickened. Slowly pour about half of the hot cream mixture into the egg yolks while whisking constantly to temper them.
Pour the tempered mixture back into the saucepan with the remaining cream. Cook over medium-low heat, stirring constantly, for 5 to 7 minutes until the custard thickens enough to coat the back of a spoon.
Remove from heat and stir in the vanilla extract. Strain the custard through a fine mesh sieve into a clean bowl to catch any bits. Press plastic wrap directly onto the surface and refrigerate until completely cold, at least 4 hours or overnight.
Pour the chilled custard into an ice cream maker and churn according to the manufacturer's instructions, usually about 20 to 25 minutes.
Transfer the churned ice cream to a freezer-safe container, press parchment paper onto the surface, and freeze for at least 4 hours until firm.
Let the ice cream sit at room temperature for 5 to 10 minutes before scooping and serving.