Better question: how to make an engaging encounter?
You make a high level encounter exciting by making it an exciting encounter to begin with, regardless of the level of the fight. Go back to the basics of encounter design, and once you have those set, customize them with some high level twists.
You can build a good high level monster fight by making it a fundamentally good fight first. The high level monster bit is just icing on the cake.
A really good resource in this regard is the AngryGM's How to Build F$&%ing Awesome Encounters!, which outlines what pieces any encounter should have, combat or no.
I'll get to his suggestions later. Below are my own.
Manage the pace properly
Your encounter has to be a roller coaster ride. Create moments of rising and falling tension while adjusing how fast you move between them. You must plan the moments when the party should lose all hope, and the moments they regain it. There should be multiple dips and rises in one encounter. You must also create the environment for them to occur naturally.
In general, your pacing curve for an encounter should look like this:
Open big with an exciting sequence. This is the party delivering their surprise round (action); or the moment the party is ambushed (suspense); the death of the princess they were escorting (sorrow); or the revelation of a dark secret that changes the PCs' perspective (tension). This is the spectacle you create to draw the players' attention.
Oscillate between high and low action in a steadily increasing manner. Follow your opening scene up with a more somber moment to set the players' baseline expectation of the action, and slowly but surely raise that baseline.
This means filling the room with less powerful minions, activating some traps that slowly get deadlier, and providing temporary shelters/reprieves that dampen the excitement of the previous sequences.
You create moments of release to give the players time to relax. Otherwise, they will acclimate to the intensity of the experience, and it will be harder to get them to engage. Crush their spirits, but give them hope, too.
Remember to follow this curve on a round-by-round and turn-by-turn basis. A really good example of this is episode 102 from Critical Role.
The BBEG delivers a Finger of Death to the cleric with the possibility of killing her (low point), but the bard counterspells it (high point), and then rolls a natural 1 (low point).
You can see the players' reactions as their spirits rise and fall in the space of one turn as the tension oscillates, and one of them even says "What is this up and down cardiac arrest of a train?"
That is what you want to hear. That is what you want to recreate. The players are so engaged, they feel like they're having a cardiac arrest.
A bit more in-depth explanation is provided by this guy on Extra Credits in the context of video games. To get the gist, begin at 0:30, watch until 1:10. To get a better idea, watch the full 6 minute video.
I won't give you the full write-up of what Angry says on his article, as you should probably read it yourself, but here's the summary:
Start with an idea, or high concept. This is something like "an encounter with a three-headed dragon guarding a princess in a tower" or "an encounter against a hydra under the water." This idea is your foundation for the whole thing.
Propose a dramatic question. These are things like: "Will the heroes save the princess from the dragon?" or "Will they escape from the collapsing tower?" This is the question about whether the heroes/PCs can accomplish their goals. It has to be a statement about what the heroes want to accomplish. When this question is answered, your encounter has ended.
Incite a call to action. In case your players don't want to answer the dramatic question, you need to pull them in with your hook. For example, you could take a whole day planning out an underground dungeon with the question in mind: "Can the heroes survive entering this dungeon?"
But when the PCs see it, they can reject your question and propose their own: "How can I destroy this dungeon without stepping foot inside it?" And there goes your entire afternoon. But if they had a call to step inside -- like a person to rescue, from a quest -- then they would have to accept your question.
Provide conflict. Now that your PCs want to achieve a goal, you have to stop them from achieving it. This is where your high level monster comes in. The monster wants to stop the heroes from achieving their goals. So if the PCs want to save the princess, the monster wants to keep her to itself. Note that putting a monster between the PCs and their goals is not a source of conflict in itself. If the monster has no reason to oppose the PCs, then there is no reason for the fight to occur.
The linked article has even more information than this. It's really long. It discusses ending an encounter, creating and intensifying "decision points", mechanical ways to implement this, and an example applying these concepts.
However, strictly speaking, you only need the four points above to create a good and complete encounter.
Making it a high level battle: just throw in the high level BBEG
As long as you follow the rising and falling pace of a good encounter, plus incorporate the basic elements of encounter building, you should come out with a gripping fight. This is very bare bones though, and the specific encounter you make will depend on the monsters and ideas you want to implement.
One high level fight I enjoyed running was against a group of five level 13 PCs against a level 20 Paladin/Sorcerer. The fight went something like this:
The players enter the room, ambushing the Sorcadin in his sleep.
They start hammering down on him. He retaliates. Damage is traded back and forth.
The PCs start going down. The healers have to bring them back up. They continue to trade damage.
One of the players start sweating, notices how imbalanced the scales are, and says this was a mistake. They should run.
They continue fighting.
The Sorcadin falls unconscious.
They don't kill him. He wakes up while they're restraining him. He casts blink with Subtle Spell and escapes.
This was the final boss fight in that campaign. They defeated him, accomplishing a specific goal the entire party shared. All the players, without being asked, said it was a great fight.
Conclusion: build a great encounter first
All that changes is the scale of the fight at higher levels, but no amount of mechanical tricks will fully replace a well-designed encounter that makes full use of our storytelling tools. Every fight, every encounter, and every campaign should follow some pacing curve that was consciously planned by the DM, to ensure maximum chances of engagement.