Grammatical function

It drew nearer, [while we sat holding our breath].

I've put your example in typical context.

I take "while" to be a preposition, and hence the bracketed expression is a preposition phrase functioning as a temporal adjunct.

For those who take "while" as a conjunction it is a subordinating one, and the expression is a finite subordinate clause.

while we sat holding our breath.

while we sat holding our breath is a free modifier. It modifies the main clause as a whole.

A free modifier differs from a bound modifier in that a bound modifier attaches to only one word or phrase.

From "ThoughtCo."

Definition:

Generally, a free modifier is a phrase or clause that modifies either the main clause or another free modifier. Phrases and clauses that can function as free modifiers include adverb phrases, adverbial clauses, participial phrases, absolute phrases, and resumptive modifiers.

What grammatical name is given to this expression?

while we sat holding our breath

what grammatical name is given to this expression"while we sat holding our breath.I think the answer is adjectivecal clause