MAKE ICE CREAM: Combine heavy cream, milk and coffee concentrate in a medium saucepan and bring to a slow simmer over medium heat. Whisk together egg yolks and sugar in a medium bowl until mixture is pale yellow and sugar has dissolved. Pour a little of the cream mixture into egg mixture to temper the eggs and whisk while adding. Keep adding cream into the yolks while whisking.
POUR mixture back into saucepan and gently cook over medium heat, stirring constantly, until mixture thickens into a custard and coats the back of a spoon, up to ten minutes. Pour thickened custard into a stainless steel bowl and set over a water bath to cool, stirring occasionally.
TRANSFER to a resealable container and refrigerate overnight. Use the custard in an to make ice cream in an ice cream maker according to the manufacturer’s instructions.
MAKE SAUCE: Combine cream, butter and sugar in a small sauce pan over medium heat and stir until sugar has dissolved and mixture begins to simmer. Whisk in chocolate pieces, then remove from the heat and continue to whisk constantly until mixture is fully combined.