The Chaperone boost exotic shotgun has been re-introduced with the Forsaken DLC. The exotic questline for this weapon is started at Amanda Holiday but is unlocked randomly. Most people report getting it after entering the Dreaming City but that is not guaranteed.
If you have the quest unlocked we are able to complete it for you.
The questline consists of the following steps:
Defeat 20 opponents in the Crucible with a shotgun
Defeat opponents in the Crucible with Arc, Solar, and Void damage.