Twice cook old bird (4)
I believe the answer is:
dodo
'old bird' is the definition.
(extinct bird)
'twice cook' is the wordplay.
'twice' says to double the letters.
'cook' becomes 'do' (do can mean to cook).
'do' duplicated is 'DODO'.
(Other definitions for dodo that I've seen before include "Ultra-conservative" , "Mauritian bird no more" , "Flightless bird, extinct since the 17th century" , "E [EXTINCT] bird" , "One deeply reactionary" .)