In a manner that is humorous, exaggerated, or mocking; in the style of a burlesque performance.
From 'burlesque' + the adverb-forming suffix '-ly.' This creates an adverb describing how something is done in a burlesquing manner.
When a Shakespearean actor performs 'burlesquely,' they're deliberately hamming it up and making the drama ridiculous—it's a wink to the audience that they know exactly how over-the-top they're being.
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