Style of European say, in cut ... (8)
I believe the answer is:
elegance
'style' is the definition.
(thesaurus)
'european say in cut' is the wordplay.
'european' becomes 'e' (abbreviation e.g. EU).
'say' becomes 'eg'.
'in' is an insertion indicator.
'cut' becomes 'lance' (I've seen this before**).
'eg' placed within 'lance' is 'legance'.
'e'+'legance'='ELEGANCE'
'of' is the link.
(Other definitions for elegance that I've seen before include "Fine quality" , "Refinement, polish" , "Graceful refinement" , "Grace and stylishness" , "Chic" .)