Here's a potential way you could try to do this, though it's quite cumbersome:
On a new Unity user, set to no greeting, maximum message length of 1 second, uncheck "Callers Can Edit Messages", and set the After Message Action to the intended user's "real" mailbox. Set up message relay on this 1-second mailbox to the user's email address. Missed Call forwarding rules on the phone will need to forward to this fake mailbox instead of the real mailbox... potentially with a *XXXXXX type of direct-to-vm CTI Route Point.
So here's the intended call flow:
Caller (5555555555) calls user's phone (123456).
User doesn't answer, phone forwards to first Unity box (546367).
With no greeting, and 1 second message time, Caller only hears initial beep before being forwarded to second Unity box (123456). The user will get an email with a subject like "Message from CALLER NAME (5555555555)" with a 1-second wav file attached at this point.
Caller gets greeting on second Unity box, and option to leave a voicemail as usual.
It's not a great solution, but if this is an absolutely-must-have-feature for a few users and you have the extra user licenses to spare, it's one potential way to achieve the effect. The email subject is a bit misleading though, as it's not actually the caller's message.
These are the paths to get to each CCX logs through CLI. They may be helpful if you are having issues accessing RTMT or downloading logs through it.
If you want to download them you have to prefix "file get " and you can add one of the options (re...