Just to chime in again, I've worked for my boss over ten years, and seen a lot of students come and go. The advice for Amy to look at the bigger picture--the completion of her degree--is good, because I feel like it's common for students (others, too!) to lose perspective and focus on immediate problems or small details, rather than seeing the big picture. While it's generally possible for students to transfer advisers, in my area it's a huge red flag and quite a mess to actually do; it's also certainly possible for a student to drop out of the program entirely, but obviously may not be what they really want.
So essentially, if Amy doesn't want to drop out or start the messy process of changing bosses, she needs to find a way to deal with her current boss professionally and focus on her goal: completing the requirements for her degree. If she's amenable to it, maybe you could help her map out what steps she still needs to complete and a rough timeline for them. Then she would have goals to aim for, and she could be more proactive about researching techniques and making a logical case for using method A rather than B, instead of getting emotional or stubborn about it.
And, not to entirely put the blame on Amy--I don't think you mentioned at what point she is in the program, but a good adviser should be keeping his student's timeline in mind as well. In my program, getting a PhD (if you already have a masters) takes about five years. After two or three years you should be done with all your classes and ready to take your preliminary exam, and after that everything is pure research and writing with a clearly-defined endpoint. Of course it happens frequently that endpoints change, but everyone should at least be aware of them.
My point is that if Amy is, say, in year four of five, of course she's getting antsy about how this thing is going to get done, and doesn't want to go off on tangential techniques that don't pan out. But if she's only in year two of five, maybe her adviser has a point that she still needs to learn a different mindset, and has time to do so. Then there's always the idea that Amy is strongly hoping to be done in four years, while her adviser has been assuming six, or something like that. Basically they need to communicate and share their ideas about her timeline.
If Amy can write out a timeline based on her own assumptions, she'll have something concrete to share with her adviser, and they can see how well their ideas match up. Then she could say, "As you can see, I feel I only have 1.5 to 2 years left, and there's a lot left to be done. I would really like to work on streamlining the research process so I have a complete story to write up in my dissertation."