Eager aunt's broken promise (9)
I believe the answer is:
guarantee
'promise' is the definition.
(guaranteeing is a kind of promising)
'eager aunt's broken' is the wordplay.
'broken' indicates an anagram.
'eager'+'aunt'='eageraunt'
'eageraunt' anagrammed gives 'GUARANTEE'.
(Other definitions for guarantee that I've seen before include "Warranty, pledge" , "security system" , "Underwrite; promise with certainty" , "Assurance, warranty" , "Something ensuring a particular outcome" .)