From my observations its obvious that the monster was feeding off of an egg it had absconded with. It is not natural for creatures to devour their own eggshell, or even the egg shells of other animals for that matter. The sole exception being the toothless snake. There's also some other points made by others I thought I would comment on and they all feed back into the monster was eating something else's egg.
First, the mind-space that Eleven is seen in (pitch black emptiness) is absolutely and clearly not the sub-dimensional shadow realm (i.e. upside-down).
Second, the sub-space dimension clearly contains reflections of its parent dimension, and those reflections occur in real-time. It would be more accurate to say impressions (made by sine wave results of destructive interference) as opposed to reflections, based on what we can see of the static matter in the sub-space. I will touch on this below, but we don't see non-electromagnetically run dynamic matter being impressed on the sub-space.
Third, electromagnetic fields have real effects on others from sub-space. (Possibly this effect works both ways, though there is little indication of this thus far).
Fourth, powerful enough emotions or thoughts (possibly amplifying EM fields) are enough to cause a localized disturbance, or even dimensional merging/rifting (as evidenced by Will's most powerful contact with the mother causing a rift/merge behind the wallpaper in an early episode).
Fifth, it is with absolute certainty that Eleven herself is not "maintaining" an entire other sub-dimension. There exist reflective elements in the sub-space of its parent dimension of which Eleven has no knowledge, and are both detailed, extensive, and expansive in nature. While we may be able to agree with Eleven's conclusion that she helped cause or even solely-caused the initial large breach, even that may be something we can still rule out.
Likely this sub-space dimension has always existed, probably from the moment the parent dimension existed.
There is nothing to suggest that this monster is the only large-scale semi-intelligent creature that exists in this other space. In fact, things would suggest quite the opposite.
Once again to re-iterate: As far as the egg is concerned, there is nothing to suggest that the egg it was feeding off of was its own birth-egg. However, in the unlikely case that it was a "Demigorgon" egg, (being that fresh) would imply that it was recently born and devouring its own shell (which is not something that any animals are known to do). However, in nature we do find creatures that eat their other's eggs, or even self-produced eggs (e.g. Hens, fish, etc). That would be a far more likely scenario.
Regardless, this affirms there is other life in the upside down, whether in the form of an egg-laying Demigorgon, or something else. This is already confirmed anyhow, as we've seen other lifeforms (e.g. Slugs) in the sub-space which are clearly not indigenous to the parent dimension.
The slugs have nothing to do with the Demigorgon's reproduction (in fact, in the original script they were going to use a Spider, but it wasn't "otherworldly" enough).
There is breathable air/oxygen in the sub-space. However, the particulate matter that permeates the air (in this area of the sub-space) is toxic. Apparently the toxicity is not fatal without extremely prolonged exposure. (Perhaps not even then)
There has been little sign of water, an arguably dynamic-matter substance. However, it could be argued that air is dynamic-matter in the same sense as well. (Or other gases and liquids). This suggests that dynamic-matter such as air or water is not impressed on the sub-space externally by the parent dimension, but rather produced internally. Just as our own Flora produce/recycle/convert our atmosphere, so too does the sub-space's own ecosystem.
Things to consider:
Considering the sub-space dimension has its own eco-system and history. (The creators have a 30+ page paper detailing how this shadow realm works, other areas, and its creatures) we may find that the demi-gorgon is the least interesting thing in it.
Where or what was the Demigorgon stealing the eggs from? Far too large for it to have laid them itself. If they are demigorgon eggs, then the adults much be terrifyingly huge.
Does Eleven attract other psychic-resonant creatures from the upside-down when she uses her telekinetic abilities? Or only her telepathic abilities? Even then, is it only when she enters her mind-space and projects herself out too greatly?