I'd never heard of ESL until your question, so I had to look that up. A local company did a lot of work developing UHT milk; I remember samples when they were first starting up production, and they were trying to find flavors additives which would mask the flavor change from the UHT processing. Since then a lot of companies have adopted the processing methods for other products, or just adopted the packaging. So, before I relied on any random box-packed product to be shelf stable, I'd check into how it was produced.
Proper UHT milk doesn't go bad until it's opened. It's sterile, and in a pretty sturdy container. Once opened, bacteria will get back into the container, and the milk will go bad as normal. When the local company started production of UHT milk, they were putting 2 year code dates on packages, and that was just the longest date the government would allow; I've had boxes 3-4 years old, which were fine.
The easiest way is just like this. What's the best approach to get runny-yolk sunny side up fried eggs? but with broken yolks and more time.
This pan would would be perfect if it had a lid that fits, but this lid will be do the trick. Your pan should be non-stick with sloping sides, 8" is perfect for 2 eggs, but larger is fine too. I set the heat to one click below medium [YMMV*], and give the burner and pan a good 2 minutes to heat up. Add the butter and let it reach the point that it stops sizzling. When the butter stops sizzling, add the eggs. To keep track for this answer, I set my timer at this point. Let them cook uncovered until the bottom is opaque, about 1 minute. Add about 1 tablespoon of water and cover.
So far everything is just like the prior answer for sunny side up eggs.
Let the eggs steam until they start to get a bit of white on top, about 1 more minute. Now poke the yolks and use a spoon to baste the top of the eggs with hot butter, replace the lid. Let them cook until they are as hard as you like them. Mine looked just about done at 3.5 minutes. Had I just let them cook one more minute, covered, they would have been fine and ready to plate.
You don't need to flip. If you do choose to flip, don't do it until they eggs are very close to being done. You can use a spatula if you like, but I much prefer the pan-flip. He demonstrates it here at 3:05. Egg Flip Just be sure that your eggs are easily sliding around before you flip. The guy in the video makes it seem harder than it is. Since your yolks are already broken, it's really super easy if you have a non-stick pan with sloping sides and plenty of butter. So I flipped at 3.5 minutes, then covered and let them go for 30 more seconds.
Pro trick: before you plate the eggs, use a paper towel to wipe any extra butter out of the pan, then just tilt the eggs out of the pan onto the plate, that makes for a tidy presentation. I never touched a spatula to these eggs.
*YMMV Your Mileage May Vary
Best Answer
The answer is a general "no", not just for eggs, but basically for any food. When you store it under the usual conditions, you already get the maximum shelf life possible.
Methods of food preservation do not magically make the food last longer, they actually produce a different food that is shelf stable (e.g. turning vegetables into pickles). Preservatives can have a role in that process, but only as one component in a specially engineered recipe.
For your broken eggs, the shelf life is 2 hours at room temperature and 3-5 days in the fridge, less if you see signs of spoilage. There is nothing that can be done to prolong that.