IMHO, the best way to avoid the awkwardness of asking an attending to make a phone call for you is to simply not do it.
The best way to make connections is to ROTATE. A phone call is not a substitute for a rotation. It will never be.
I'm pretty confident I made a very good impression on several of the attendings that I worked with during my rotations. I asked those that I felt would back me most strongly to write me a letter of rec.
For all I know, nobody made a single call on my behalf.
But at multiple interviews, I was asked what I had done to make such a positive impression on the attendings I worked with. So I can only assume the letters expressed very strong support.
If I had wanted to go the phone-call route, I am sure I would only have asked my home Chairman (you don't really want to rotate for 4 weeks at a program just to turn around and ask them to help you get in somewhere else, do you?) and even then only if he had brought it up or said to me that he wanted to help me get into a particular program. And even then, it would only be if there was ONE program in particular that I really knew I wanted to rank #1 and that I also knew he had connections at.
I'm happy with how I went about it. I think trying to arrange a phone call would've unnecessarily complicated matters and maybe could've backfired (if, say the Chairman wanted me to stay at my home program). Of course, match day is next week, so time will tell...
