Vegetable a girl cooked at first without hesitation (8)
I believe the answer is:
celeriac
'vegetable' is the definition.
(celeriac is a kind of vegetable)
'a girl cooked at first without hesitation' is the wordplay.
'a girl' becomes 'celia' (**).
'at first' says to take the initial letters.
'without' means one lot of letters goes inside another ('without' can be similar in meaning to 'outside').
'hesitation' becomes 'er'.
The initial letter of 'cooked' is 'c'.
'celia'+'c'='celiac'
'celiac' enclosing 'er' is 'CELERIAC'.
(Other definitions for celeriac that I've seen before include "It has a root like a turnip" , "Plant with turnip-like root" , "Odd root vegetable for Alec Rice" , "Clear ice (anag) - it is used as a vegetable" , "Turnip-rooted celery" .)