If “the Divine Essence is simply the abstract definition of what God is” then why do the Orthodox claim we cannot behold the Essence of God? I behold abstract definitions all the time – not much to them.
Because, in the Byzantine approach, to know the Essence of something is to comprehend it. The Essence, in this approach, is taken as a whole; the issue isn’t the abstractness, but the infinity of this nature. In Latin theology this is approached by saying that God’s Essence can be known, but not comprehended, and this is possibly because of a different nuance in what Essence means, and what knowing means. Latin theological language has more flexibility in this matter.
If you mean that the Essence is our definition of what God is, then how is this “is” different than the “being” you say is part of the Energy?
This is where the definition “super-essential” comes into play. In all other things, the essence is something other than their being, and the essence is prior to the being, and the Byzantine approach takes this as the model for the language. In God’s case, God’s being, which in created things is subsequent to the essence, is the reason for the essence. At the same time, however, energy follows from essence by definition, so there is a conundrum when it comes to God. Therefore God is said to be “super-essential” because His Essence follows from Him being.
Again this is different from the Latin theological approach, though both address the same issue and come to the same basic conclusion in their own manners.
I have heard Orthodox claim that the Essence and the Energy are both God – then in the case of the Energy, how can it be God without it being Who He is (Essence)?
The Divine Energy is, to use Latin terminology a bit, “God being God”. The Divine Energy can further be “broken down” to individual concepts, which themselves are the same in God but distinguished by us, such as Mercy and Justice and Living, ect. These are not the “super-essential Essence” because none of them contain the wholeness of God in the human mind, and because the “super-essential Essence” can’t be comprehended. The Energy is not “Who He is” because the Energy is not wholly transcendent, whereas transcendence is part of “Who He is”. To use a crude and limited example, when you touch my arm you truly touch me, but you do not truly touch my mind or will or soul by touching my arm with yours despite the fact that those things are indeed truly me as well. This example is crude and weak, however, because God is simple, while we are composite, but if you consider the mind and such as transcending the arm, and not merely being another part of me, the example might be more beneficial.
You are using the Latin theological language and approach with your question, which is perfectly valid (and in some respects more nuanced and preferred, IMO, though the Byzantine is better in other respects), but it doesn’t help in understanding what is meant by the Byzantine usage. In the Latin terminology the Essence encapsulates everything that is God, while in the Byzantine the term is less all-encompassing.
You’ve lost me again. How can a definition be God? My confusion wants to ask the silly question of how many other gods are in my Webster dictionary?

Definitions themselves aren’t God, but rather Divine Essence is the “definition of God”. If the infinity of God could be summed up in a Webster’s Dictionary entry, the Divine Essence would be what you would read when you looked up “God”. Now, the definition of, say, “deer” in Webster’s is not the wholeness of what it means to be a deer, because the definition itself is not actualized in a real deer being a deer.
Of course there is no mingling of the natures. But, if the Orthodox are correct and our nature cannot behold God’s Essence (absolutely and without exception), then how could Jesus’ human nature behold His divine nature? How could the two natures be hypostatically united if one can’t look upon the other?
They are hypostatically united because one Person encompasses and utilizes both natures. Similar to how we could say that our soul and body are hypostatically united, and our soul touches our body but our body does not touch our soul. Christ’s human nature, in this theological approach, does not behold the Divine Nature, any more than His eyes could look at His own soul. Again, in the Latin approach it is said that Christ’s human nature always beheld the Divine Essence, precisely because Divine Essence is used slightly differently, and includes what is called the Divine Energy in Byzantine theology.
Hope that helps!
Peace and God bless!