I read about using mayo awhile back and decided to try it.
Honestly, I'm unimpressed. I've tried multiple heat settings, different pans (cast iron and nonstick), preheating and not preheating the pan, and melting butter into the pan beforehand.
None of those provided me with a better grilled cheese. All of them were worse or the same as when I used butter.
So either I'm a really shitty food scientist or it's not all it's cracked up to be. For the record, I'm guessing that the latter is most likely the cause.
What I did start doing however, is to begin with toasting the inside. If you haven't been toasting the inside of your grilled cheese, you're totally missing out. See J Kenji Lopez-Alt's approach in the Food Lab for the best grilled cheese recipe