My first guess is that a DTMF or a "restart call" signal is coming from the switch which causes Connection and Unity to restart the greeting being played - call flow traces or port status monitor output may point to some further clues here - not much to work with here...
i saw something similar a couple of months ago, as Jeff says there is little info but maybe this helps
the scenario was Unity 5 using latest TSP and some calls would hear the greeting twice also. the config was that calls were coming from a PBX with QSIG link to CUCM, they had path replacement on the config enabled and that caused the TSP to detect the connection change and to start again the greeting.
as i mentioned, maybe not your exact scenario but it's worth a shot
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