The phrase "once being asked" is correct and usable in written English.
You can use it when you want to reference a question that has already been asked or to indicate that something happened after a particular question was posed. For example, "Once being asked if the project was complete, she realized the task was not finished."
I remember her once being asked out of the blue, at a private lunch, what she thought about Mozambique.
Last week I entered and left the UK three times without once being asked for identification or a passport.
It is not an overstatement to say that economics students can go through their entire degree without once being asked to express an opinion.
(Funny story: I remember once being asked by a public relations person where their App review was in my review queue. Amazing, considering I don't have an iPhone and never agreed to any App review in the first place!).
Journalist Paul Carr emails us to say that he was able to upload his book, Bringing Nothing To The Party: Confessions Of A New Media Whore, to the Kindle Store " in less than 24 hrs without even once being asked to prove that a) I was the author b) I had the electronic rights for the US".
Rushing them to the vets, learning how to administer subcutaneous fluids and insulin shots, and even once, being asked to euthanize a pet, have all befallen me during my sits.
Mick Jagger once was asked, 'What makes a hit song?
He had once been asked why he chose to pursue baseball instead of hockey.
The great Rosabeth Kanter of Harvard Business School once was asked what men could do to advance women's leadership, and she replied: "The laundry".
Mr. Paulson once was asked about Cigar's ownership: Didn't the horse actually belong to his wife, as earlier listings showed?
Dr. Linn Goldberg of Portland, Ore., said he had once been asked by parents to provide steroids so that their son might get a football scholarship to Notre Dame.
