A quiet look by doctor shows self-possession (6)
I believe the answer is:
aplomb
'self-possession' is the definition.
(synonyms)
'a quiet look by doctor' is the wordplay.
'quiet' becomes 'p' (music symbol for 'piano', ie soft or quiet).
'look' becomes 'lo' (archaic exclamation meaning 'look!').
'by' is a charade indicator (letters next to each other).
'doctor' becomes 'mb' (Bachelor of Medicine, Medicinae Baccalaureus).
'a'+'p'+'lo'+'mb'='APLOMB'
'shows' is the link.
(Other definitions for aplomb that I've seen before include "Self-possession - coolness" , "Self-satisfied assurance" , "Cool self-assurance" , "coolness is required" , "Air of self-satisfied confidence" .)