Although employees are often encouraged to engage in self-promotion on the job, it sounds like your employee took this advice too literally. Her unilateral upgrading of her title is one hundred percent unacceptable.
Assuming that she is correct and her work is now at a director level, her next step should have been to discuss this with you. The fact that she anointed herself as director is a critical event that raises questions about her judgment across the board. It should have been intuitively obvious to her that this type of a change is not a great idea.
Your first step is to meet with her and reaffirm that she is still a manager. You should also express your concern that she would make this type of change without any discussion or approvals. Be sure to give her an opportunity to discuss her rationale for this action. You are sure to learn more about her, and you may also learn more about your company.
The irony in this situation is that her decision to bestow the title of "director" upon herself has clearly reduced the likelihood of your bestowing such a title on her.
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