I learned this from someone at a cocktail party and it has literally changed my life forever, or at least the grilled cheese making part of my life.
The explanation is that the oil in the mayonnaise will fry the sandwich and the proteins in the egg will crisp up to create the perfect diner crunch. This is why you have to use real mayo. If you mistakenly use Miracle Whip or other food products made by the good folks at Exxon then your sandwich will more closely resemble the cleanup tools from the gulf oil spill than anything likely to come out of Flo's Diner.
I'm not going to bother with ingredients or instructions as if you need them then you probably shouldn't be using a stove or a knife.