The method is complicated and nobody fully understands it except for Mr. Benedict and, of course, Curtain himself. What we know of it is that, after a modification by Mr. Curtain, one needn't sit in The Whisperer to be brainswept by it and the brainsweeping "range" of the Whisperer. In addition, brainsweeping does not technically delete memories, but rather "sweep" them out of sight and out of mind, like dust under a rug.
This method was used on Milligan by Mr. Curtain, erasing all of his memories except for the phrase "mill... again", a memory of his last time with his daughter Kate, who asked if they could return to a certain mill-pond. The phrase, in his mind, became "Milligan" and he took it as his name. To this day we don't know Milligan's maiden name.
Other victims of brainsweeping are:
Along with other students (known as "special recruits"), graduates, and the Helpers.