According to the OED, tout court means in short, in little, simply, without qualification or addition.
But tout itself means to peep, peer, look out; to gaze.
So my question is how can “peep court” mean “in short”?
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The entire phrase is French. As an adjective, tout means “all” In French, but as an adverb as it is here, tout means “quite”. It’s modifying the adjective court, which means “short”. The longer OED entry for the adverbial sense reads in part:
By the way, the OED has seven completely different entries for the word tout; in other words, it has seven different tout words. Your “peep” definition just doesn’t apply to this one.
With the meanings the OED gives it in its separate entry, tout is an English verb, rhyming with 'spout'. In the expression tout court it's a French adjective, rhyming, in an English pronunciation, with 'you'.
EDIT: A French adverb, not adjective.