I am a long-time native speaker of English and have never heard the phrase "broken question". I understand it to mean a question that asks how to do something that one should not be doing in the first place, or that is posed in such a way that the actual answer to the question is made off-limits.
When playing with matches, should one light one match at a time, or several?
If you can't take the train or a bus, because you have no cash on you at the moment, is it better to hitch a ride with a sociopath, or to borrow a police car while the cop is in the donut shop?
I have always understood the supposedly paradoxical question, What is the sound of one hand clapping? to be not a paradox at all, but a simple example of a broken question. We can easily go astray philosophically by posing our questions in a way that shuts the door on truth.