Doctor Dread beginning to rub part of chest (6)
I believe the answer is:
drawer
'part of chest' is the definition.
('drawer' is part of 'chest')
'doctor dread beginning to rub' is the wordplay.
'doctor' becomes 'dr' (abbreviation).
'dread' becomes 'awe' (awe can mean a feeling of dread).
'beginning to' indicates taking the first letters.
The initial letter of 'rub' is 'r'.
'dr'+'awe'+'r'='DRAWER'
(Other definitions for drawer that I've seen before include "Cheque signatory" , "Artist" , "Reward (anag.)" , "Part of tallboy" , "Sliding part in a piece of furniture" .)