Half a menu hides celebrity (4)

I believe the answer is:

fame

'celebrity' is the definition.
(celebrity can mean the quality of being famous)

'half a menu hides' is the wordplay.
'hides' indicates a hidden word.
'FAME' can be found hidden inside 'half a menu'.

(Other definitions for fame that I've seen before include "Musical - stardom" , "Note" , "Repute" , "Notoriety or celebrity" , "Prominence" .)

I've seen this clue in The Mirror.

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