In a small saucepan, heat the heavy cream over medium heat.
Add the instant espresso, salt, and vanilla extract and whisk to combine.
Bring to a simmer while stirring.
Keep stirring while the heavy cream begins to boil.
In a mixing bowl, whisk the eggs and the powdered Erythritol until light and fluffy.
Bring a saucepan of water to a boil (to make a double boiler) and place the bowl with the egg mixture on top.
Gradually add the heavy cream mixture to the egg mixture while stirring constantly for 5 minutes until the mixture thickens up. Add the gin or vodka.
Allow the mixture to chill in the fridge for at least 30 minutes. Transfer into a freezer-proof container and place in the freezer.
Stir the mixture every hour for the first two hours to achieve a creamy consistency.
Serve frozen.