Carried by old cook, starter of fillet coming up, meat dish (4,10)
I believe the answer is:
beef wellington
'meat dish' is the definition.
(I've seen this before)
'carried by old cook starter of fillet coming up' is the wordplay.
'carried by' is an insertion indicator.
'old cook' becomes 'beeton' (I've seen this before).
'starter of' indicates taking the first letters.
'coming up' becomes 'welling' (welling is a kind of coming up).
The first letter of 'fillet' is 'f'.
'f'+'welling'='fwelling'
'beeton' enclosing 'fwelling' is 'BEEF WELLINGTON'.
(Other definitions for beef wellington that I've seen before include "meat dish" , "dish - with 10a" , "meal" , "Meat-in-pastry dish" , "Menu item" .)