In my Collins COBUILD Advanced Learner’s Dictionary 5th Edition, they wrote:
An engaging person or thing is pleasant, interesting, and entertaining.
…one of her most engaging and least known novels…
He was engaging company.
I wonder whether we should use ‘a’ before ‘engaging’ or not? Because I often see sentences like these:
She is a charming dancer.
He is a hard-working engineer.
In your examples, “person”, “dancer” and “engineer” are all singular countable nouns, so they require articles. This is not affected by the intervening adjectives “engaging”, “charming” and “hard-working”. For example, just as we say “She is a dancer”, we say “She is a charming dancer”, “She is a beautiful dancer”, and so on for any other adjective.
In “He was engaging company”, the noun “company” is uncountable, so an article is not required.