Quiet? Notice that herring-like fish? (4)
I believe the answer is:
'herring-like fish?' is the definition.
(I've seen this before)
'quiet? notice' is the wordplay.
'quiet?' becomes 'sh' (both can be said to tell someone to be quiet).
'notice' becomes 'ad' (short for advert).
'that' acts as a link.
(Other definitions for shad that I've seen before include "Herring-like fish", "Fish related to the herring", "fish that's been caught", "Fish; dash (anag.)".)