Put the milk, sugar and chocolate into a saucepan and heat gently until the chocolate has melted and the sugar dissolved.
Pour the warm mixture over the beaten eggs, stirring constantly.
Return the mixture to the pan and cook over low heat, stirring constantly, until the custard begins to thicken slightly.
Strain the mixture into a bowl and add the vanilla extract.
Let it cool.
Whip the cream until it forms soft peaks, then whisk it into the cooled custard.
Stir in the chocolate chips.
Turn the mixture into a freezer container, cover and freeze until firm.
About 30 minutes before serving transfer the ice cream to the refrigerator to soften.
Spoon or scoop the ice cream into individual dishes and serve with chocolate cookies, if liked.