Composure shown by doctor after a quiet look (6)
I believe the answer is:
aplomb
'composure' is the definition.
(aplomb is a kind of composure)
'doctor after a quiet look' is the wordplay.
'doctor' becomes 'mb' (Bachelor of Medicine, Medicinae Baccalaureus).
'after' says to put letters next to each other.
'quiet' becomes 'p' (music symbol for 'piano', ie soft or quiet).
'look' becomes 'lo' ('lo' is an archaic exclamation meaning 'look!').
'a'+'p'+'lo'='aplo'
'mb' after 'aplo' is 'APLOMB'.
'shown by' is the link.
(Other definitions for aplomb that I've seen before include "Self-satisfied assurance" , "Air of self-satisfied confidence" , "Cool self-assurance" , "Self-possession, assurance" , "Great coolness and composure under strain" .)