The chicken is a fatheaded bird (4)

I believe the answer is:

fowl

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'the chicken' is the definition.
(I've seen this before)

'fatheaded bird' is the wordplay.
'fatheaded' becomes 'f' (this might be a standard abbreviation I've not previously seen).
'bird' becomes 'owl' (owl is a kind of bird).
'f'+'owl'='FOWL'

'is a' is the link.

Can you help me to learn more?

(Other definitions for fowl that I've seen before include "eg hen" , "eg Chicken" , "Bird to be eaten or hunted as game" , "Domesticated bird" , "Bird kept for eggs and meat" .)

I've seen this clue in the Evening Standard.

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