Eggcorn

[Wilson Gray]

The American Heritage Dictionary of
Idioms<http://www.answers.com/topic/cut-to-the-chase>has the following
about
*_cut to the chase_*:

Get to the point, get on with it, as in *We don't have time to go into that,
so let's cut to the chase*. This usage alludes to editing (cutting) film so
as to get to the exciting chase scene in a motion picture. [Slang; 1920s]

I can only guess that the variant *_cut to the cheese*_ relies on a similar
idea, only related to food: a cheese course is usually served at the end of
a meal.
 What about a reanalysis of _cut the cheese_ = "fart," influenced by "cut to
the chase"?

-Wilson Gray

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The American Dialect Society - http://www.americandialect.org