Terribly chilly at Mayor's place? (4,4)
I believe the answer is:
city hall
'mayor'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 anagramming the letters.
'chilly'+'at'='chillyat'
'chillyat' with letters rearranged gives '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" .)