What can I do to regrow a burnt eyebrow that is not growing back?

If the "burnt" eyebrow has not grown back within one year, it probably will never grow back.  One option may be hair transplant.

The burn is a scar tissue and it is almost impossible to grow hair on a scarred tissue. The best way to regrow your eyebrow is hair transplant

The burn must be deep in order to kill all hair follicles in the area. If it was that deep, then it is likely that you would have needed a surgical intervention to repair the burnt area. If it healed spontaneously without significant scarring, then there still may be a chance that hair grows back. I would wait atleast 6 months to draw any conclusions. If nothing grows back then you can consider an eyebrow transplant.
Eyebrow transplants are very sophisticated procedures and are more difficult than hair transplants. The fact that there is a difference between the texture and biology of scalp hairs and eyebrow hairs makes it more difficult to achieve a natural result. A handful of surgeons in the world are capable of producing such natural results so you have to be very careful in choosing one. You are going to have a permanent change in your facial appearance, which means the decision is very critical. From the many subtleties of eyebrow transplants, I would like to mention one here. That is FUSS (the strip method) is superior to FUE when doing eyebrow transplants, because it allows for preparing long-haired grafts which allows for a much better control of angle, direction and especially curvature.