How can I find a PhD position to work on a highly specific idea of my own?
You should look for faculty working in the field that you are interested in researching, and then pitch your research idea directly to them.

Make sure that you do a good job with a preliminary review of the literature for your idea and understand the professor's specific research interests, so that your letter does not sound like any of the hundreds of mail-merge messages that professors get every week.

If your idea is really worthwhile and you do a good targeted job of pitching it, changes are that someone will agree to work with you on it.  However, the reality is that most students come into a PhD without a really good sense of existing literature or the feasibility of their ideas.  It is for this reason that there is great value in being open-minded about your PhD thesis.

Many students make the mistake of thinking that the PhD should be the pinnacle of their career, solving some grand problem in human existence.  This might happen for a few figures in history, but, more typically, the PhD is the start of a research career and not the end of it.