Vegetable piece shortly cut (7)
I believe the answer is:
parsnip
'vegetable' is the definition.
(parsnip is a kind of vegetable)
'shortly cut' is the wordplay.
'shortly' becomes 'pars' (I can't explain this - if you can you should give a lot more credence to this answer).
'cut' becomes 'nip' (nipping is a kind of cutting).
'pars'+'nip'='PARSNIP'
'piece' is the link.
(Other definitions for parsnip that I've seen before include "Pale carrot-like vegetable" , "Sweet root vegetable" , "A white root vegetable" , "Yellow root vegetable" , "food" .)