[Kazeto]

Engineering officers get experience when you craft or disassemble things. But it is really slow unless you are being a scrap hound.

Alternatively, you could just take all of your officers for a field trip on some planet (or for multiple field trips). That gives them all experience for killing things, and is probably the quickest way of getting it if you are willing to risk it (although, with how numerous your away team is then, it requires messing up really badly for your officers to die).