Although your project seems pretty interesting, I must admit that it'll be quite challenging and probably require several weeks of work.
(To my knowledge) There haven't been any project/guide that does something similar to this. You will have to program audio mirroring from your phone to your Edison, meaning that you will probably have to program a custom app for your phone and for your Edison. You will also have to install the audio drivers for your headphones and that will probably mean that you’ll have to modify the Yocto image.
I found this link online that talks about how to stream music from an Android phone to a PC over Wi-Fi. I know that you are interested in streaming phone calls but it might give you a place to start.