2020-07-15 last update. Too long didn't read; FDEV killed the egg with fleet carriers patch 3, what you can read below is basiscally history now, sorry 
What is The Egg?
It is mining related.
You are in Col 285 Sector Cc-k a38-2 body 1 mining low temperature diamonds (LTD) and in system chat commanders (CMDR) keep talking about "egg", what is it?
This egg is four sub surface deposit low temperature diamond asteroid in specific location, it also has very low rotation/spin so it is easy to mine. It also has high percentage of laser mining content.
These type asteroids are often referred as "4 SSD" where CMDRs short cut the Sub Surface Deposit, now obviously everyone knows SSD means Solid State Drive (a computer device to store data).
Why Does It Matter?
Four sub surface deposits of LTDs will get you on a good day 90 tons of low temperature diamonds, just from one asteroid (this, the egg asteroid in question, not sure about every other they might have different yields).
The yield depends of how good miner you are, how nicely you hit all the blue bars in the drilling mini-game process. But even a terrible miner should get 50t out of the egg.
Because you can find this asteroid very easily, this is awesome location to start your mining session... or re-spawn it (read below).
HOWTO Find The Egg?
Its somewhat complicated to explain; drop into body 1 ring on the triple overlapping hotspot for LTDs, specifically into the "middle one". Middle one is relative as there are no up/down directions in space heh, when you face the planet and 3 LTD hotspots, its the one on the left (assuming you fly directly from the main star without rolling your ship, as said, its complicated).
Once you have dropped into the hotspot, fly above the hotspot AND the ring asteroids. You do not need to fly INTO the asteroid field itself to get close to the hotspot marker. You are roughly 3 - 4 km above the marker when you arrive close to the asteroid field.
Now turn your ship nose to point into the planet, its relative but assuming you are still not rolling your ship, its right edge of the planet put your ships mouse widget over that (you do fly it on, do you?).
Now fly ahead, boost if you got it. Around 15km or so (cant actually remember) you see a puff of white smoke/thing on the left side, it seems to be always there I have never seen it not to be. Soon after you see same thing in front / below of you. Around 22km or so stop boosting, ie be prepared to stop. When distance to hotspot reaches 24.5km cut throttle and pitch nose down, hit pulse wave analyzer and the bright orange/yellow asteroid in front of you is the EGG
Yes I know, its kind of complicated to explain without images as up/down does not exist in space, its easy to orient your ship in a wrong way that these directions above do not work (they are reversed, sort of).
Re-Spawn The Egg
CMDR's always ask how to re-spawn the sub surface deposits in the asteroid. Yes you can do that. If you move just slightly more than 25km away from the asteroid, turn back and fly back to it, all sub surface deposits have respawned but not the laser mining content which stays depleted (not sure how long, some CMDRs say 2hrs, some say until client restart).
Self Launched Fighter Exploit
There is/was(?) an game exploit which allows you to refresh / re-spawn the sub surface deposits in asteroids by using self launched fighter (SLF) switching.
You need a fighter bay in your ship, this is class 5 module I believe. You do NOT need to hire a crew for the SLF, in fact you absolutely should not hire because they take 10% cut from any mineral sales lateron.
Fly to the asteroid, mine it fully until its depleted. Turn of your ships thrusters. Get into SLF, fly it above the asteroid field until about 25km or so. Turn off SLF thrusters. Switch back to mothership, switch to SLF (yes again. this you only need to do once, the first time) and finally switch one more time back to mothership (yes, odd). Shoot new prospector limpet into the asteroid and you find the laser mining content depleted but sub surface deposits have been re-spawn.
Turn on your ship thrusters and mine the asteroid, once done... just repeat the above until your cargo hold is full of diamonds or you run out of sub surface deposit missiles 
Note: after you have done this loop, you do not need to switch back and forth twice, just swith to SLF, wait for 5 seconds or so, then switch back to mothership and asteroid has been respawned. No need to switch back and forth twice.
2020-06-27 Frontier Developments confirmed this is a bug and exploit which will be fixed in the future patch. They did try to fix it in a patch, but CMDRs are still reporting that SLF asteroid re-spawn works.
Note: this SLF exploit has nothing to do with the "egg", but is often associated with it. You can do similar fly away then come back and SLF switch-eroo to any asteroid.
2 Hour Timeout
Some CMDRs in system chat claim there is 2 hour timeout when you leave the instance, meaning you jump into super cruise or was it hyper jump away from the system not sure, that when you come back the re-spawn of this asteroid will not work for two hours.
For me personally on 2020-07-07 there was no such issue, I mined the asteroid, flew 25km away to the hotspot marker and back, several times, then hyper jumped to different system into my carrier, shut down Elite client completely during my lunch. Then started Elite back up and it was not even an hour later when I went back to the asteroid and all the sub surface deposits were there, I did not see any evidence of this "2hrs timeout" other CMDRs claim. This 2hrs timer is not present with SLF exploit either. So dunno?
Disclaimer - This Is Shady
PMC believes all this asteroid "mapping", re-spawning and SLF exploiting is just that, shady way of exploiting the game. David Braben did not design mining to be such stupid re-spawning of asteroids, its completely opposite what mining should be. This is unfortunately how Elite "cobra" game engine works with procedural generation of system bodies, ring asteroids in this case.
It is very sad to see people use such methods or even the SLF exploit to make easy credits. But hey we aren't judging, everyone plays Elite the way they choose to. I don't want anyone to come telling me how can I play the game so I won't be saying that to anyone either.
It is up to each CMDR to decide for themselves if they want to conduct such behavior. Seems like most people in july 2020 are into mining not because they like mining itself but because LTDs give such insane amounts of credits for fairly low amount of hours spent playing.
PMC mines because we love mining, and sure yes the credits ain't bad either 
See also PMC Elite Dangerous: Mining.