In a medium saucepan, combine the sugar, cocoa powder, and salt. Stir in the heavy cream and milk and cook over medium heat, stirring constantly, until the mixture is hot but not boiling.
Remove from the heat and stir in the espresso powder and vanilla extract. Let cool to room temperature.
Refrigerate for at least 4 hours or overnight.
Freeze in an ice cream maker according to the manufacturer's instructions.
Serve immediately or transfer to an airtight container and freeze for up to 1 week.