My wife is eligible for H-4 EAD, I know there is no formal way for expediting the process but I've seen a clause in F-1 (OPT) EAD process called "financial loss" and if you've got a job and submit the offer letter while you're waiting for an EAD to get approved, they process your case faster and you get the EAD much sooner than expected. Does this rule apply for H4-EAD as well?
See clip from Attorney Rajiv S. Khanna's conference call video that addresses this question.
https://youtu.be/y6RtaKhuoIY?t=864
FAQ Transcript:
I do not know. But let me tell you this when you look at the expedite criteria for any government document they are basically the same. If you can show government error or economic necessity or emergency or some kind of humanitarian considerations - there is five or six criteria then apply. As general observation I think it should be possible to expedite H-4 EAD if you meet the requirements. Theoretically it is possible to expedite the H-4 EAD.