Chief having broken eggs leaves (7)
I believe the answer is:
romaine
'leaves' is the definition.
(type of lettuce, a leafy vegetable)
'chief having broken eggs' is the wordplay.
'chief' becomes 'main' (synonyms).
'having broken' indicates putting letters inside (inserted letters break into the word).
'eggs' becomes 'roe' (fish eggs).
'main' put inside 'roe' is 'ROMAINE'.
(Other definitions for romaine that I've seen before include "lettuce in America" , "Variety of lettuce" , "Type of lettuce - I arm one (anag)" , "Cos lettuce" , "lettuce in the US" .)