In Paris, miss classy type of Madame, in a manner of speaking (6,7)

'a manner of speaking' is the definition.
Although both the answer and definition are singular nouns, I can't see how they can define each other.

'in paris miss classy type of madame' is the wordplay.
I cannot quite see how this works, but
'miss' could be 'ache' (aching is a kind of missing) and 'ache' is found in the answer.
This may be the basis of clue (or it may be nonsense).

'in' acts as a link.

Can you help me to learn more?