Tease Brussels’s elite primarily about posh woman driver (10)

'woman driver' is the definition.
(I know this)

'tease brussels's elite primarily about posh' is the wordplay.
'tease' becomes 'chaff' (chaff can mean to tease or mock).
'brussels's' becomes 'EU's' (European Parliament is based in Brussels).
'primarily' indicates taking the first letters.
'about' is an insertion indicator.
'posh' becomes 'u' (abbreviation for upper-class - opposite to non-U).
The first letter of 'elite' is 'e'.
'chaff'+'eus'+'e'='chaffeuse'
'chaffeuse' placed around 'u' is 'CHAUFFEUSE'.