Cutter is dreadfully chesty (6)
I believe the answer is:
scythe
'cutter' is the definition.
(scythe is a kind of cutter)
'dreadfully chesty' is the wordplay.
'dreadfully' indicates an anagram (a dreadful spelling).
'chesty' is an anagram of 'SCYTHE'.
'is' is the link.
(Other definitions for scythe that I've seen before include "Manual hay-cutter wielded with both hands" , "Cut down - grass-cutting implement" , "Crop-cutting blade" , "Reaping blade" , "cut down the weed?" .)