Crossword Genius

Picked up composer’s Asian cocktail (3,3)

Ross

I believe the answer is:

mai tai

I'm a little stuck... Click here to teach me more about this clue!

'cocktail' is the definition.
(I've seen this before)

'picked up composer's asian' is the wordplay.
I cannot really understand how this works, but
'asian' could be 'tai' (Tai is an example) and 'tai' is present in the answer.
The remaining letters 'mai' is a valid word which might be clued in a way I don't see.
This may be the basis of the clue (or it may be nonsense).

Can you help me to learn more?

(Other definitions for mai tai that I've seen before include "Cocktail" , "drink?" .)

I've seen this clue in the Sydney Morning Herald.
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