Been looking over chef cutting up right plant (5,4)

'plant' is the definition.
(beech fern is a kind of plant)

'been looking over chef cutting up right' is the wordplay.
I cannot really see how this works, but
an anagram of 'chef' is 'echf' which is present in the answer.
'right' could be 'r' (common abbreviation) and 'r' is present in the answer.
The remaining letters 'been' is a valid word which might be clued in a way I don't understand.
This may be the basis of clue (or it may be nonsense).

Can you help me to learn more?