Beat the egg yolks, sugar and brown sugar in a medium bowl just until blended.
Heat the cream in a saucepan until almost boiling. Pour the cream in a thin stream into the egg yolk mixture, whisking constantly. Return the mixture to the saucepan and cook over medium heat to make a light custard, about 5-7 minutes. Do not allow to boil. Remove from heat.
Dissolve the instant coffee in the espresso. Stir in the creme de cacao, vanilla and cinnamon and whisk into the custard. Refrigerate covered until cold. Freeze in ice cream maker.
