Bosses regularly filling bigwig's drink quickly (4,3)

I believe the answer is:

toss off

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

'drink quickly' is the definition.
'toss off' can be an answer for 'drink' (tossing off is a kind of drinking). I am unsure of the 'quickly' bit.

'bosses regularly filling bigwig's' is the wordplay.
'regularly' indicates alternate letters (regularly take one letter, leave next etc.).
'filling' is an insertion indicator.
'bigwig's' becomes 'toff' (I can't justify this - if you can you should believe this answer much more).
The alternating letters of 'bosses' are 'oss'.
'oss' placed into 'toff' is 'TOSS OFF'.

Can you help me to learn more?

(Other definitions for toss off that I've seen before include "Accomplish easily" , "dispense with the pre-match routine" .)

I've seen this clue in The Guardian.

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