Bill’s the judge (4)
I believe the answer is:
beak
I believe this is a double definition.
'bill's' is the first definition.
('beak' can be a synonym of 'bill')
'the judge' is the second definition.
(thesaurus)
(Other definitions for beak that I've seen before include "Bird's bill" , "Judge - bill" , "Bird bill; magistrate" , "Bill; nose" , "Bird's mouth" .)