in a broad way vs at large

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

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

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

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

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