Profs are busy and it is unlikely that they will give you one-on-one advice if they don't have any stakes in your project.
Therefore, you must offer something in return. I have not tried this myself, but it supposedly works (Gray 2010, p. 60-61): Volunteer to read and comment on a paper, now or in the future, from the person whose advice you are soliciting. Even very senior researchers need feedback, even from juniors. Why? Because problems of clarity and organization in a late draft are more obvious to non-experts, and because non-experts can more easily find overriding (but hidden) themes in an early draft. In your inquiry, address the following:
- How their work has informed yours. Ask specific questions aimed at the intersections of their work and yours
- Which sections and issues you need feedback on
- Whether you want hints to additional reading and citations
- Explain that you want just a "quick" read (not an editorial read)
- Ask when you might expect to hear back from them
- Promise acknowledgements
Moreover, to solicit comments on my PhD project from people besides my advisors I mostly relied on conferences and smaller workshops:
- I applied for workshops and conferences outside my institution. Many workshops and conferences in political science, such as this one, cater specifically to PhD students.
- We had a weekly colloquium at my PhD institution where I presented mostly every six months. This was a great opportunity to discuss halfway finished work like draft chapters.
- Students from my PhD institutions invited external speakers to participate in a lecture series. Often the speakers were willing to hold a small workshop the following day (before flying home) where a handful of PhD students could discuss their work and ask for advice. If a lecture series with externals exists at your institutions, maybe you can suggest to add a workshop.
- You might also be able to propose a thematic workshop (on particular methods or theories), depending on the demand at your institution and the available funding (if external guests are needed). It is best to determine the demand first, i.e. to ask around who would attend such a workshop, before proposing this to your superiors.
- At my workplace, the younger researchers established a workshop with a particular "discussant" format. In each session (bi-weekly), someone presented and discussed the paper of another participant. It was expected that the papers not be too polished and do still have many loose ends so there could be a lively discussion how to proceed from the work. The informal atmosphere (coffee and rolls!) and the discussant model made this highly productive, although mostly junior faculty participated. Again, this is something that you could try to establish at your institution.
Admittedly, all these proposals rely on written contributions of probably more than a two-page memo. However, also your thesis needs to be more than two pages at some point. So why not start to write a draft chapter right away? If you want useful feedback, you have to provide tangible input. Depending on the venue, the contribution can be anywhere from tentative "think-piece" (5.) to almost final draft (1.). You mention that you are doing field-work right now. Why not polish your field-notes and elaborate how your preliminary findings form the field can inform your theory/research question and which new challenges you have encountered?
BTW: I would be interested in hearing form senior faculty if they have ever encountered an inquiry for advice from external PhD students and if they would respond positively if offered a reading in return.
Gray, Tara 2010: Publish and Flourish, Ashland: BookMasters.