at large vs in a broad way

at large is an adjective but in a broad way is not an adjective.

Word	Noun	Adjective	Verb	Adverb
at large	No	Yes	No	Yes
in a broad way	No	No	No	Yes

As adverbs, at large and in a broad way are synonyms defined as:

at large (adjective)	in a broad way (adjective)
having escaped, especially from confinement

at large (adverb)	in a broad way (adverb)
in a general fashion	in a general fashion