A quiet learner with old doctor showing composure (6)
I believe the answer is:
aplomb
'composure' is the definition.
(aplomb is a kind of composure)
'a quiet learner with old doctor' is the wordplay.
'quiet' becomes 'p' (music symbol for 'piano').
'learner' becomes 'l'.
'with' says to put letters next to each other.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'doctor' becomes 'mb' (Bachelor of Medicine, Medicinae Baccalaureus**).
'a'+'p'+'l'+'o'+'mb'='APLOMB'
'showing' acts as a link.
(Other definitions for aplomb that I've seen before include "Self confidence" , "Self-satisfied dignity and confidence" , "Self-satisfied poise" , "Self-possession - coolness" , "Cool self-assurance" .)