Monster grabs rook and chicken part (6)
I believe the answer is:
breast
'chicken part' is the definition.
('breast' is part of 'chicken')
'monster grabs rook' is the wordplay.
'monster' becomes 'beast' (I've seen this in another clue).
'grabs' indicates putting letters inside.
'rook' becomes 'R' (chess abbreviation).
'beast' enclosing 'r' is 'BREAST'.
'and' is the link.
(Other definitions for breast that I've seen before include "Barest (anag.)" , "seat of emotions" , "Outer thorax" , "Milk producer" , "White chicken meat" .)