Chef barely notices dessert going early? (4,5)

I believe the answer is:

head start

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'chef barely notices dessert going early?' is the definition.
I can't judge whether this defines the answer.

'chef barely notices dessert' is the wordplay.
'barely' says to take the centre (the word is bare of its external letters).
'notices' becomes 'ads' (advertisements).
'dessert' becomes 'tart' (type of dessert).
The centre of 'chef' is 'he'.
'he'+'ads'+'tart'='HEAD START'

Can you help me to learn more?

(Other definitions for head start that I've seen before include "Advantage gained by beginning early" , "An early advantage for the star ad" , "advantage, initially" , "Boost?" , "'The advantage of beginning ahead of others (4,5)'" .)

I've seen this clue in The Independent.

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