What is the best way to feel out a friend to find out if he or she has any doubts or criticisms?
Great question!
I agree with Cofty that the safest course is to say nothing at all.
But if you want to get through to your friend and are willing to take risks, I would recommend listening a lot and saying little. If the friend expresses any doubts or questions, DON'T jump in and express your doubts or try to show him what you have discovered... at least not right away. That could overwhelm him, scare him off, and get you DFd. Instead, be nonchalant and noncommittal about it so he knows he has a safe place to vent.
If you want to initiate discussion, I think the best way is to ask your friend what he thought about the latest WT "new light" article, such as the latest articles on the "faithful and discreet slave." You might mention that you just wondered because you personally have been having a little trouble grasping all the ramifications of the new teaching, probably because you had been trained in the "old light" and even taught it for years.
They might use that as an opportunity to open up about their own doubts or criticisms.
But if they are shocked by your comments or seem uncomfortable by your comment, you can say, "Well, it's no big deal. I'm just trying to understand things better and it's taking time."
If things go wrong, your friend might get the elders checking on you. You can assure them that you are working through the issue in your own mind, trying to understand the various changes. As long as you don't directly contradict the WT or bash the organization, most likely they will let it go at that. But "more likely" isn't a guarantee. You ARE taking risks by saying anything that might be construed as sowing discord or apostasy.
Of course, if the elders have been making "shepherding calls" on you or hounding you already because they know you have doubts, you need to be extremely cautious because they may be looking for any chance to catch you in something for which they can DF you.