crowing vs cock-a-hoop

crowing is a noun but cock-a-hoop is not a noun.

Word	Noun	Adjective	Verb	Adverb
crowing	Yes	Yes	No	No
cock-a-hoop	No	Yes	No	No

As adjectives, crowing and cock-a-hoop are synonyms defined as:

Other synonyms of crowing include big, boastful, braggart, bragging, braggy, self-aggrandising, self-aggrandizing.

crowing (noun)	cock-a-hoop (noun)
an instance of boastful talk

crowing (adjective)	cock-a-hoop (adjective)
exhibiting self-importance	exhibiting self-importance