Two ways of doing an egg

One of my favorite ways of preparing eggs is to heat up a nonstick pan, do a quick wipe with butter, and then crack an egg on the surface. Put on the lid, and bring down the heat. So-called "steam frying", this gives me many of the benefits of sunny side up, without actually having to futz with the egg, yolk comes out runny, I use a minimal amount of fat, and there's that nice crispy edge.

Then, as I read John Besh's book My New Orleans, he describes something called Eggs on a plate, which is essentially the same thing, with a difference - he starts with a cold pan, greases it, and puts the egg on it, before he heats it. I tried this method, and it resulted in a rather intriguing texture, more akin to that stage between soft boiled and hard boiled eggdom. However, the edge was unpleasantly tough rather than browned and crispy, and I suspect that if I worked in a restaurant, I'd trim that off.