In a manner that broils, burns, or creates intense heat; in a way characterized by conflict or heated tension.
From 'broil' + -ing + -ly, creating an adverb that describes actions done in a broiling or heated manner. This is a rare, archaic formation.
This word is so rarely used that it's mostly found in old literature—it captures the idea of intensity and heat in a way that's become redundant with modern English.
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