Medic consumed to a higher degree (9)
I believe the answer is:
doctorate
'a higher degree' is the definition.
(I've seen this before)
'medic consumed' is the wordplay.
'medic' becomes 'doctor' (I've seen this before).
'consumed' becomes 'ate' (to eat is to consume food).
'doctor'+'ate'='DOCTORATE'
'to' acts as a link.
(Other definitions for doctorate that I've seen before include "Higher degree" , "High degree" , "Highest academic degree" , "Postgraduate degree" , "Academic qualification" .)