Cooked (it's British-grown) (7)
I believe the answer is:
braised
'cooked' is the definition.
(similar in meaning)
'british-grown' is the wordplay.
'british' becomes 'b' (abbreviation e.g. in BBC).
'grown' becomes 'raised' ('raise' can be a synonym of 'grow').
'b'+'raised'='BRAISED'
'it's' acts as a link.
(Other definitions for braised that I've seen before include "Stewed in closed pot" , "Fried then slowly stewed" .)