The mien of a first-class ruling head (3)
I believe the answer is:
air
'the mien' is the definition.
(thesaurus)
'a first-class ruling head' is the wordplay.
'a first class' becomes 'ai' (A1 - top quality).
'head' suggests taking the first letters.
The first letter of 'ruling' is 'r'.
'ai'+'r'='AIR'
'of' acts as a link.
(Other definitions for air that I've seen before include "Bearing; tune" , "voice giving > impression" , "See 21" , "Sky" , "Atmosphere surrounding us" .)