The Black Tower was started by Larien, a male channeler from Cairhien and the first Asha'man.  While not the Dragon Reborn, he still thought that isolating men who could channel until they either went mad or were gentled by Aes Sedai was likely to lead to further disasters.  

Instead of going the route of a False Dragon, Larien began to quietly find other male channelers, and they began to learn from each other.  Very carefully, their numbers grew, and they began to organize.  They congregated in the west of Andor where Larien laid out his plan for their training.

The place they would build would be the Black Tower, as opposed to the White Tower.  This would be their place of safety, and it would be their place of power, power they would take back from the Dark One.  

Larien established the black uniforms of the Asha'man as well as the ranks, with a mind toward defending the world against the Dark One and showing that men who could channel would not be tossed out like trash.  The sword would symbolize the weapon their abilities were against the Shadow, and the dragon pin was to symbolize the creature they were all feared to be.  They all knew his plan was dangerous, but with care and consideration, and the other options being what they were, they all decided the risk was worth it.

As they expected, they were found out.  The Whitecloaks attacked first, but were run off by their numbers and power.  The Aes Sedai hovered, knowing that fighting a group of male channelers at that size was suicide, but they also needed to be watched.

Ten years after Larien's death, a powerful Asha'man name Iradal, a scholar from Arad Doman, gathered a group of trusted men with the two access keys of the Choedan Kal and not much of a plan past that.  Using their contacts around the smaller towns of Andor, they began to look for female channelers who did not wear the shawl to work with them to cleanse Saidin of the taint.

After years of persuasion, they managed to find a group of three wise women in the area who admitted to channeling without training.  They were strong, to have lasted this long, and were very wary of linking with men.  Still, with the goal of cleansing Saidin, they agreed that the risk was worth it.  If they could rid the world of that, they decided that any sacrifice in the fallout would be worth it.

Iradal linked with the oldest and strongest of the wise women, a woman name Rallie, and they executed his plan.  They carefully selected an isolated spot in the Mountains of Mist to execute their plan.  They made sure no trade routes were around, and that no one was in sight, and they began, their companions standing guard.  

For hours, Saidar was used to sift out the taint of Saidin.  Rallie doubled over from the effort of trying to contain and channel as much power as she did, but tried to stay awake and conscious for as long as she could.  Iradal was consumed in his work and his theory, hurling the raw power into the natural barriers of the Mountains.

When they were done, Rallie was barely alive, and when the power finally winked out from her access key, so did her ability to touch it.  Iradal was clinging to consciousness, but his companions congratulated him on a mission accomplished.  A nearly circular crater was left all the way through the section of the Mountains of the Mist, called the Breach in the aftermath.  One of the wise women who had helped shield from falling rubble took the access keys and swore to hide them where no one would find them.  In the year 679 NE, the Dark One's taint was cleansed from Saidin, and the Black Tower began to take in numbers they had never dreamed of.
Abby Soulane - Accepted
Torianin Soulane - Aes Sedai of the Green Ajah
Jhanic of the Serai Tomanelle - Algai d'siswai and Asha'man