Crossword Genius

Quiet provider of milkshake (7)

Ross

I believe the answer is:

shudder

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'milkshake' is the definition.
Although both the answer and definition are singular nouns, I don't understand how they can define each other.

'quiet provider' is the wordplay.
'quiet' becomes 'sh' (both can be said to tell someone to be quiet).
'provider' becomes 'udder' (I can't justify this - if you can you should believe this answer much more).
'sh'+'udder'='SHUDDER'

'of' acts as a link.

Can you help me to learn more?

(Other definitions for shudder that I've seen before include "Convulse" , "Shake, quaver" , "Vibration" , "Tremble through fear" , "Shiver with horror" .)

I've seen this clue in The Guardian.
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