Flan rapidly cut by that man (6)
I believe the answer is:
quiche
'flan' is the definition.
(type of flan)
'rapidly cut by that man' is the wordplay.
'rapidly' becomes 'quick'.
'cut' means to remove the last letter.
'by' says to put letters next to each other.
'that man' becomes 'he' ('he' refers to a man).
'quick' with its last letter taken away is 'quic'.
'quic'+'he'='QUICHE'
(Other definitions for quiche that I've seen before include "Savoury egg-based flan" , "Open tart with egg filling" , "Type of savoury tart" , "Open tart with savoury egg filing" , "Tart filled with savoury ingredients" .)