How to add water/milk to the sunny side up eggs?

While I've never tried adding water to the skillet, as Josh Caswell suggests, I would stay away from adding milk or cream. I've never heard of anyone adding water or milk to eggs that aren't beaten, and it makes sense; the point of adding the extra liquid is to change the consistency of the beaten eggs and make them fluffier.

Regardless, if you do try this, do it with water and not milk. When I was a kid and first trying to learn how to make eggs, I sometimes added far too much milk to them. Even with beating the milk into the eggs before cooking, if there was too much milk it woulds start to pool on the bottom of the skillet, and I'd wind up with burnt milk froth along with my eggs. I suspect the sugars in the milk burn much more quickly than the eggs cook, but whatever the explanation, the result is unappetizing. Adding milk to unbeaten eggs would almost surely result in something similar - or you'd wind up with a puddle of warm milk and an un- or under-cooked egg.

I'm not quite clear on the reason you want to do this, but you can certainly add a small amount of water (or veg stock) to the skillet -- enough to just cover the bottom -- right before you put the egg in. (I have in fact done this many times.)

The result is that the edges and bottom of the whites will not be fried, but will have a soft, almost coddled, consistency by the time the yolk sets up. You should still heat the skillet as if you are going to cook it in fat, but don't add the fat (except perhaps butter, which is, of course, partially water). You may find it more difficult to get the egg out of the pan once it's done (cast iron has always worked best for me), and when I've done this, I've finished it with a ring mold or pastry cutter for presentation.

It's also best to use the freshest eggs you possibly can, because as far as I can see, the water's lower temp (compared to oil) will allow them to spread more before setting if they're old.

The quote in your question is clearly aimed at eggs that are mixed (ie omelettes, scrambled eggs etc) before cooking, not for whole eggs sunny side up. The benefits of adding more water is to slow down the cooking of those mixed eggs.

If you add a liquid, water or milk, you'll be poaching the eggs rather than frying them in fat or oil as most sunny side up eggs are cooked.

If poaching is what you want to do, there's plenty of instructions on how to do this out on the web. Poached eggs are equally as good if not nicer than fried 'sunny side up' ones in my opinion.

You don't - for sunny-side up eggs. Mixing in water, milk or cream only alters the eggs when it can be fully incorporated into the eggs as part of a mixture - so, for scrambled eggs, where it's all mixed into a homogeneous mixture.

With any kind of fried egg, where the egg is intact, there's no way you can get the liquid to mix with the eggs while leaving it physically distinct as yolk and whites, so, really, trying to wouldn't add anything to the eggs. Nor would you want to, for sunny-side up or any of the other varieties of intact fried eggs (over-easy is my favorite), in my opinion.

I would shy away from adding milk/cream to the pan when cooking the eggs but I do often use a bit of water and a lid to promote steam in the pan to help set the top of the egg when cooking sunny side up. I think you could run the risk of scorching and creating a mess with dairy but I believe the idea of the directions you encountered was to promote steam so that you don't have a really fried/hard bottom of the egg and a top that is too runny and undercooked.

One maybe-possibility is this: I once saw a recipe where the egg whites were beaten into peaks, scooped onto a hot frying pan and a shallow well made in the center, and the reserved yolk slid into the well to cook with the whites to make something that was essentially a fried egg (no extra ingredients), but with a vastly different texture to the whites.

So, if you really wanted, you might be able to separate your eggs, whip the egg whites, adding a (very) little milk or cream as you prefer, and re-assembling your egg when frying. The texture of the whites would be very fluffy and soft, and the yolk would tend to be less done for being added halfway through the cooking (and possibly having extra volume between it and the heat).

The thing is, the milk or cream won't mix well with egg whites unless you really mix it hard enough to change the texture, so you probably would have to mix it frothy and thin to get the milk to mix, then whip it thickly enough to hold together and not run all over the pan. You would likely only be able to use a spoonful or so of liquid in an egg if you don't want to over-thin your whipped egg whites, or make the structure loose enough to deflate. And it would be a lot of effort, for every fried egg you make. On the other hand, no one but you can say whether or not it's worth the effort.