Place a large mixing bowl in the freezer.

In another large bowl, whisk together the egg yolks and sugar until thick and pale yellow. In a saucepan bring the half-and-half to a boil. Gradually pour into the egg mixture, whisking until the sugar dissolves. Let stand 5 minutes and then chill.

Stir the vanilla, cold espresso and Kahlua into the chilled custard. Pass through a fine strainer. Pour into an ice cream maker and process according to the manufacturer's instructions. When the ice cream is nearly done, melt the chocolate over simmering water.

Transfer the frozen ice cream into the chilled bowl from the freezer. Pour in the melted chocolate, stirring and mixing vigorously with a wooden spoon. Transfer to a container and freeze until firm.