How can I hire a professor to answer my advanced technical questions in a niche area? In short, How does one go about hiring a professor or someone really well established in their field for a private tutoring session?

I tried looking online but can’t find much, mainly websites for tutors that are practitioners or “experts” as in with with n years of experience at best. This doesn’t cut it at my level as I am a researcher with a PhD and already have published papers on my topic. 

However, I am very passionate about my subject and always looking to grow. So I do have questions, very technical (in my case computer science) and would happily pay a fee for consulting with a professor in the field.

Any advice?
Unless you are in a place where professors are greatly underpaid and generally abused, this seems like a quest that is bound to fail. People are busy and they aren't normally driven by money (other than large pots of grant money). 

What I think you want, instead, is a collaboration. That can be started with just an email, provided that you have something to contribute yourself. This requires you to do a bit of research to find someone with common interests to form a two-way partnership. 

Perhaps the story of Srinivasa Ramanujan and his relationship with Hardy will be instructive.