Crossword Genius

Terribly chilly at Mayor of London's place (4,4)

Ross

I believe the answer is:

city hall

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'mayor of london's place' is the definition.
Although both the answer and definition are singular nouns, I can't see how they can define each other.

'terribly chilly at' is the wordplay.
'terribly' indicates an anagram.
'chilly'+'at'='chillyat'
'chillyat' is an anagram of 'CITY HALL'.

Can you help me to learn more?

(Other definitions for city hall that I've seen before include "local government building" , "Local authority" , "In the US, municipal offices collectively" , "US municipal government building" .)

I've seen this clue in The Independent.
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