Closer to being a hush, say? (7)

I believe the answer is:

shutter

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'closer' is the definition.

'hush say?' is the wordplay.
'hush' becomes 'sh'.
'say?' becomes 'utter' (saying is a kind of uttering).
'sh'+'utter'='SHUTTER'

'to being a' acts as a link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.

Can you help me to learn more?

(Other definitions for shutter that I've seen before include "Hinged window cover" , "Camera mechanism" , "Removable cover" , "application for Windows?" , "Hinged window blind" .)

I've seen this clue in the Evening Standard.

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