Removing hair except for top bit? That's cutting (5)

I believe the answer is:

axing

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'that's cutting' is the definition.
(I've seen this before)

I cannot understand how the rest of the clue works.

(Other definitions for axing that I've seen before include "Chopping, wood or jobs say" , "Cutting down" , "Dismissing (employees)" , "Reducing expenditure sharply" , "Sacking" .)

I've seen this clue in The Irish Times.

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