The ideal sunny side up should have a bright creamy yolk, and smooth, evenly cooked egg whites. Slimy egg whites, and rubbery egg whites are deal breakers for me.
I tried three different cooking methods, and tried each method twice; once with oil, and once with unsalted butter. I found that the eggs cooked in oil tended to be more rubbery, I will chalk that up to the chef’s error, and inability to control the temperature properly.
The first method involves dropping the egg into a medium heat pan with lots of oil/butter, letting the egg start to sizzle on the edges, then spooning the heated oil over the whites of the eggs till they look cooked. I actually tried this one three times, but could not get it right. Usually, the whites came out with a rubbery consistency, and were full of bubbles, the yolk was cooked more than I would like, and two of the three times some of the hot oil got onto the yolk and caused bubbles to form on top.
The second method involves dropping the egg into a low-medium heat pan with some oil/butter, then letting it cook very slowly uncovered. Both trials of this egg came out quite nicely, with a runny yolk, and mostly cooked egg whites. However, the base of the yolk was slightly more cooked and congealed than I would have liked.
The final method involves dropping the egg into a medium heat pan with a little bit of oil/butter, then once the base of the egg whites start to turn white move the egg around the pan by gently turning, or lightly shaking the pan over the heat for one-two minutes, cover the pan, remove it from heat, and continue gently turning the pan. When the egg whites look cooked (one-two minutes, depending on the heat of the pan), remove the lid and serve. Both trials of this one came out perfectly; no liquidy egg whites, a beautifully runny yolk, and no bubbles or rubberiness to the eggs. The one downside was that the whites of the egg were more spread out when cooked this way, but were also cooked much more evenly.
I don’t know the science behind it all, but if I had to guess, I’d say that the movement of the latter method allows the egg to cook more evenly by avoiding any one part of the egg from sitting on a hotspot of the pan, and the covering allows the top of the egg whites to steam evenly instead of pouring hot oil over the egg.