Big Eddy’s cautious, beady eyes (listening to me) (9)
I believe the answer is:
charybdis
'big eddy's' is the definition.
(whirlpool or eddy monster of Greek myth)
'cautious beady eyes listening to me' is the wordplay.
I cannot really understand how this works, but
'cautious' could be 'chary' (similar in meaning) and 'chary' is found in the answer.
'me' could be 'i' and 'i' is located in the answer.
This may be the basis of clue (or it may be nonsense).
(Other definitions for charybdis that I've seen before include "Whirlpool opposite Scylla" , "Whirlpool, across from 14" , "Mythical whirlpool" , "Scylla's counterpart" , "monster" .)