"You'll never walk again, but you will fly." With those words the Three-Eyed Raven helped launch one of Game of Thrones' greatest theories: Brandon Stark will warg into a dragon and help the living defeat the army of the dead. Even if he's not powerful enough to overcome the might and magic of a dragon right now, his other powers could help him learn how to do so.
Brandon Stark's abilities make him arguably the most powerful person in the world. His ability to warg into another creature and control their mind—which makes him a skinchanger or beastling—is a rare skill. On top of that, he has greensight. His "green dreams" give him prophetic visions of things that will come to pass. Like most prophecies, they are difficult to discern until after the events happen. Someone who is both a warg and has "the Sight" is known as a greenseer.
These powers come from the Children of the Forest. It's believed the first greenseers were the wise leaders of that ancient race. Only one in a thousand Children are skinchangers, and only one in a thousand of them also have greensight. The odds of Bran, a human born thousands of years after the Children fled north, being a greenseer are even more remote.
The First Men believed the greenseers carved faces in weirwoods so they could see through the trees, maybe through time itself to the past and future. It's why they cut so many down during their war with the Children. From what we know about Bran's and the original Three-Eyed Raven's powers, the First Men were correct. Jojen Reed, who only has greensight, tells Bran in A Storm of Swords the greatest of the greenseers "could look through the eyes of the weirwoods" to "see the truth that lies beneath the world." We know Bran can do that. He has found hidden truths in the past, like Jon's birth at the Tower of Joy, and he's had visions of what would happen in the future, like when he saw Cersei blowing up the Sept of Baelor with wildfire before it happened.
But it might be something else Jojen said that could be our best clue about whether Bran can warg into a dragon. Jojen told him the most powerful greenseers "could wear the skins of any beast that flies or swims or crawls." Dragons definitely qualify as beasts that fly.
Not all skinchangers' abilities are equal. A normal warg can only enter and control the mind of a dog or wolf, but the most powerful skinchangers can control far bigger creatures. We've seen wargs enter the minds of eagles, owls, and ravens in the TV series, but in the books, a wildling known as Varamyr Sixskins controls three wolves, an eagle, a snow bear, and a shadowcat—the last two being much larger animals.
While some skinchangers can warg into another human, it's considered a monstrosity. An injured, desperate Varamyr tries to do that anyway in A Dance of Dragons when the White Walkers are nearby. It doesn't work when the woman whose body he tries to enter goes mad, biting off her tongue and ripping her eyes out, to make him leave. Bran, on the other hand, wargs into Hodor with ease; it's unclear if that's because Hodor is simple-minded or willingly lets Bran control him during moments of crisis. Or maybe Bran is just that strong. No matter the answer, Bran's other amazing greenseer ability to use a weirwood to see through time points to his warg skills being formidable—or maybe the better word is destructive, since using those two powers together led to him destroying young Wylis/Hodor's mind.
Those abilities are all impressive, but can any warg, even if he is a great greenseer, control a dragon?
No record, nor a tall tale, of anyone doing that exists—dragons didn't exist until after the Children of the Forest had mostly disappeared from the world. No one knows for certain where the first dragons came from. The legend varies from place to place. In Qarth they say a second moon split open and dragon eggs fell out. In Asshai they say the great beasts came from Beyond the Shadow Lands and they tamed them. But the greatest dragonlords the world has ever known, the Valyrians, said they found them in volcanoes.
The Valyrians used magic to control their dragons, thought to be magical creatures themselves. Magic in the world decreased when the dragons went extinct; summers got shorter while winters grew longer. The Valyrians' spells let them build the greatest empire the world has ever known, but their methods were lost to time after the Doom of Valyria.
Daenerys proves it's still possible to ride a dragon without magic. Dragons will sometimes accept someone with Valyrian blood ("the blood of the dragon") as their dragonrider, though being of Valyrian descent doesn't guarantee they will accept you.
If a dragon accepts you, it is an exclusive bond and no one else can ride them. If a dragonrider climbs on the back of a different dragon, even if that dragon knows them and has a different dragonrider, they will throw the person off. And dragonriders can only mount their own dragon so long as their dragon lives. Only when one of them dies can either a dragon or rider make another connection. Dragons seem to know instinctively if their rider dies. They don't have to be physically near them when they pass.
(Note: a dragon will allow their dragonrider to bring others along for a spin. That's how Daenerys was able to rescue the survivors of the Frozen Lake.)
Just because you are a dragonrider doesn't mean your dragon will always obey you. "A dragon is not a slave," Daenerys says. The Mother of Dragons has been unable to control her children sometimes, including her own mount Drogon.
There's no guarantee Bran—who is not the blood of the dragon—can control one. When you warg into any creature you control them, and dragons are not slaves. If he enters one of their minds, he could accidentally turn them into a chaotic weapon of death.
Dragons are still creatures however, and Bran might be the most powerful warg who has ever lived. That alone might be enough to wear the skin of a beast as mighty as a dragon.
The Valyrians controlled their dragons with magic that has been lost, but nothing is lost to Bran who can see through time. Whether the secrets to taming dragons is in Asshai or in the annals of Valyria's Freehold, he can access them. He could use warging, taming a dragon with spells, or a combination of those abilities to establish control.
And we have other reasons to think he will.
Benjen Stark told his nephew that when the Night King finds his way to the world of men, Bran "will be there waiting for him" and he "will be ready." Jojen once told Bran that when the White Walkers come "the Night’s Watch can’t stop them, nor can the kings of Westeros and all their armies," because as Jojen also said, "The only one thing that matters" is Bran...That might be because Bran will be there to greet the the Night King, not on the ground in his own body, but in the sky as the greatest weapon the world has ever known, in the skin of a dragon.
"You'll never walk again, but you will fly," the Three-Eyed Raven told Bran, and he has. But he possesses the skills to do it like no greenseer ever has.
Of course, that raises an important question: can Bran warg into the Night King's dragon?
For that answer you'll have to wait for part two.