I have an employee is will be conducting a training class. Performing this duty is not part of her job description. Can I pay her as a 1099 employee for the "billed" amount of the training?

Yes.

Well I am not sure, why are they doing this training, who normally does this training.
If he is acting within the scope of being your employee, you determine when and where, how long and what the pay is, I am not sure they are not just an employee doing additional duties.

For example it not my job or within my job to drive the semi truck, but during a strike I have had to drive the truck past the front gate because drivers would not cross the picket line.

So did they offer to provde this service as a independent contractor ?

And why would you not merely pay them their hourly rate to do this job ?
Or do they want additional money to do it.