Bird westerners originally precooked, unexpectedly (10)

I believe the answer is:

woodpecker

'bird' is the definition.
(woodpecker is a kind of bird)

'westerners originally precooked unexpectedly' is the wordplay.
'originally' suggests taking the first letters.
'unexpectedly' is an anagram indicator.
The initial letter of 'westerners' is 'w'.
'precooked' is an anagram of 'oodpecker'.
'w'+'oodpecker'='WOODPECKER'

(Other definitions for woodpecker that I've seen before include "Knocking bird" , "Tapping bird" , "Bird that drills holes" .)

I've seen this clue in The Times.

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