How to pick the brain of a professor with whom you have no connection I want to learn more about a specific part of a narrow, fledgling field. I want to have a 30 minute conversation with a the burgeoning experts in this area of study -- they are all researchers at universities. 

I have completed my undergrad in a related field, but at a different institution, and without formal training in the specialized area I'm investigating. But I have read up enough to be able to ask informed questions.

How can I get these busy people to give me a half an hour (or more) of their life when I don't know anyone in their network and am an outsider? What can I offer them in exchange for their time? What is a good way to make contact and avoid a totally "cold call"?
You are right that it is difficult to make contact with a busy expert when you have no network contacts to draw upon. One of the best ways to make contact is to attend a conference that covers your area of interest; this can be pricey, but in my field the major conference even has "Meet the Experts" lunches for student attendees. At the conference, register for pre-meeting, student, and/or special interest group events that may have a either a career development focus or are likely to have a more specialized attendance and thus more opportunities to approach individuals with whom you'd like to talk.

While some experts may, in fact, respond to a "cold call" (or cold email, as the case may be), the fact is that most are probably putting out fires and dealing with higher-level responsibilities, and may simply be too busy. In that case you may want to try moving down the ladder a bit...a graduate student or postdoc in their lab may be a bit more available to chat, and may actually have good advice for you about the next step in your career, as well as information about the nitty-gritty of the specialized area their lab works on. 

My best advice is just go out there and make your best effort to get in touch, provided you convey respect for their time and withdraw gracefully if they aren't able to make time for you. Some investigators may respond very favorably--you might be surprised! Nothing ventured, nothing gained...