two and barbecue other. in the phrase go two and barbecue, to be defeated in the first two games of a double-elimination tournament. (source: Double-Tongued Dictionary)
Further reading
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Perhaps an extension of the single-elimination version one and done?
Many recipes for BBQ involve cooking the meat twice. For example steaming the meat first to tenderize it before further “flavor cooking” over a flame.
Perhaps that could have something to do with its origin.