In a manner of forgetting or showing an act of forgetting; while engaged in forgetting.
From 'forgetting' (present participle of 'forget') plus '-ly' adverbial suffix. This creates an adverb from the gerund/participle form rather than from an adjective, a less common but valid formation in English.
This is an extremely rare word, almost never seen in modern English, but it's grammatically valid. It shows how English allows speakers to create new words by combining existing pieces—most new combinations never catch on, but they're 'correct' according to our system of word formation.
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