This phrase means that someone is being prim and proper with a cool kind of demeanour. But from what event or phenomenon or occurrence was this idiom derived from, and when?
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It goes on to say,
Eric Partridge offers this citation in A Dictionary of Slang and Unconventional English:
So, onward to cheese, we find — nothing useful. Partridge is curiously silent on the subject, at least as it relates to this case directly. I wonder if his reticence might be due to one unsavory meaning of the word, use of cheese for smegma, giving the whole thing a lewd sexual connotation. But I can find nothing in the O.E.D. or elsewhere to support this contention. Still, if cheese is somehow related to butter in this context, this could well shed a whole new light on that phrase as well. But given the lack of sources, I have to conclude this is a dead end.
Still, we might also consider another meaning of butter as "fulsome flattery, unctuous praise" (Partridge). A young woman who was demure, as Partridge calls it, would be likely to receive much of this kind of butter.
A woman who was "reserved, modest, and shy" would probably seem a little cold, and certainly resistant to the kind of buttering flattery that might come her way.
I would offer the suggestion that butter and cheese though both dairy products have varying melting points. Though butter may go soft at room temperature (and some cheeses) I might infer that the suggestion from this quote may be that cheese would in fact melt in her mouth and therefore butter wouldn't stand a chance. The key to this being that she 'looks' as though butter wouldn't melt although the truth is that cheese would not choke her. Perhaps therefore it has 'melted' or at least gone soft. Good old hot lips.