Audacity of English cook enthralling foremost European (10)
I believe the answer is:
effrontery
'audacity' is the definition.
(I know that effrontery is a type of audacity)
'english cook enthralling foremost european' is the wordplay.
'english' becomes 'E' (abbreviation as in OED).
'cook' becomes 'fry' (fry cook is a kind of cook).
'enthralling' means one lot of letters goes inside another.
'foremost' becomes 'front' (similar in meaning).
'european' becomes 'E' (abbreviation e.g. EU).
'front'+'e'='fronte'
'fry' going around 'fronte' is 'ffrontery'.
'e'+'ffrontery'='EFFRONTERY'
'of' is the link.
(Other definitions for effrontery that I've seen before include "Impudence" , "Arrogant presumption" , "Impertinence" , "Audacity" , "lots of cheek!" .)