Those runny parts in sunny-side up eggs.
The South Beach Diet got me into eating a lot more eggs than I've eaten in years. I prefer omelets and sunny-side-up. Cooking omelets is no problem, but I just can't seem to figure out a way to completely cook out the clear (slimy) stuff in the sunny-side-ups. I use a gas range, on medium heat, and use non-stick pans. I sometimes will cover the pan, towards the end, and that helps a bit, but also causes that white substance to form over the yolks. Oh.... and I break the whites with a spatula as the eggs cook, so that more direct heat can cook them. Any help appreciated.
If you're frying those eggs in butter, you can always use a small spoon to baste the egg whites with the hot butter from the pan. Yum.
Covering the pan is one way to cook the white, and yes it will cover the yolk. Nothing wrong with that. If you want the eggs to cook faster, put the cover on the (empty) pan while it is heating.
If you are cooking non stick with out butter/oil, you can add a small amount of water to the pan, this will create steam, steam will help cook the top of the egg.
No need to "break the whites" as the egg cooks.
Another way to cook the white is to cook ~1 lb of bacon and use the fat. Cook the egg(s) in the bacon fat, once the egg sets, spoon hot bacon fat over the entire egg.
I saw a TV show, where the cook discarded some of the white.
I think you've solved your own problem. Sunny side up eggs have a little bit of eggwhite over the yolk. When you cover the pan, "that white substance" forming on the yolk is the eggwhite cooking from the extra heat. If you don't like raw white on your egg, you can flip the egg to cook it, but then you won't have a "sunny side up" egg, you'll have an "over easy" egg, which sounds like what you want. This is fairly easily done, but the yolk can break, and if it does, you are entirely within your rights as chef to curse loudly and throw things around the kitchen without regard to potential damage.....
Alternatively, you can soft boil your eggs - 4 minutes or slightly less in boiling water will give you completely cooked whites with a still-runny-yolk. It was my grandmother's favorite way to eat eggs, and mine too. This also gives you an excuse to buy cool looking egg cups and an "egg-top-chopper-offer" scissors-like-device made in the shape of a chicken.
Covering a sunny-side up egg so that the white cooks over the loose yolk is officially called a "basted" egg, I belive (or at least that is what my dad always called it!)
If you like the yolks runny, but want the white cooked, then you should use the lid.
I melt the butter over med-high heat in my nonstick pan, put in my 1 or 2 eggs, then cover for 60 sec. Then I check it. If the white is still not done, on goes the lid for another 25-30 sec at a time - keep your eye on it till the white is set, and enjoy!
Actually, properly fried eggs are *way* better than what passes for fried eggs in the US. You just need to use a deep enough pot and use wooden -- *not* metal or plastic -- utensils (eggs fried at that temperature will stick to metal or plastic, don't ask me why). I use a small, but deeper than skillet-depth, pot. It's easier to do the eggs one at a time, and drain them on a little rack. I've used a variety of vegetable-based oils; all are very nice, though olive offers the most flavor and viscosity (but with a slightly higher risk of overheating the oil past the smoking point). The edges of the white get all frilly and overall a much nicer texture than US-style frying.