The Italian Coffee Queue (ICQ) models two different customers sending jobs to a queue and competing with one another to have their jobs served first. Jobs arrive at a given Poisson rate. Each job can either wait at the tail of the queue or else jump ahead to tag along waiting jobs belonging to the same class. This comes at a fee paid to the competing customer. On the other hand, the server incurs a switching cost when serving two consecutive jobs belonging to different classes. It follows that the server's switching cost is minimized when all jobs of both classes jump the queue - whenever possible - w.p.1. We formulate the game between the two customers and we provide necessary and sufficient conditions under which the Nash equilibrium between the customers coincides with the server's optimal operating point.