Nice lady cooked cakes in the morning (6)

I believe the answer is:

madame

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'nice lady' is the definition.
(title for a woman in France eg in Nice)

I cannot understand how the rest of the clue works.

Can you help me to learn more?

(Other definitions for madame that I've seen before include "A French-speaking woman" , "Married woman in Montpellier?" , "Married woman abroad" , "Title for a Frenchwoman" , "Frenchwoman's title" .)

I've seen this clue in The Independent.

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