Greedy eater served with one monstrous bird (6)
I believe the answer is:
pigeon
'bird' is the definition.
(pigeon is a kind of bird)
'greedy eater served with one monstrous' is the wordplay.
'greedy eater' becomes 'pig' (I've seen this before).
'served with' is a charade indicator (letters next to each other).
'monstrous' indicates anagramming the letters.
'one' anagrammed gives 'eon'.
'pig'+'eon'='PIGEON'
(Other definitions for pigeon that I've seen before include "Trafalgar Square bird?" , "Epigon (anag.)" , "birdie" , "See 5" , "Common dove" .)