Engineering extension courses were a series of classes offered by Starfleet Medical Academy. They were designed to give students basic information about technology to supplement their medical training. The courses included information on starship operations.
In 2370, Julian Bashir told Miles O'Brien about the engineering extension courses he took while they were hiding from Kellerun forces on T'Lani III. He offered to assist O'Brien with repairing a communications system, but O'Brien declined. (DS9: "Armageddon Game")
In 2371, Bashir, while experiencing an elaborate comatose delusion, was able to use his engineering skills to reactivate Deep Space 9's central computer network by rerouting the primary command processor and bypass disabled systems. (DS9: "Distant Voices")
Later that year, the changeling who impersonated Ambassador Krajensky and Julian Bashir claimed as Bashir that he was putting his extension courses to work connecting a new diagnostic console when confronted by O'Brien in a Jefferies tube on board the USS Defiant. (DS9: "The Adversary")