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. 

Hi, I would probably suggest using a great healthy vitamin for times of stress. Such as Vivascal. I would also recommend trying Latisse. We have seen pretty good results with that. Also, I would refrain from over tweezing and waxing. 

Three months is still not long enough to allow a new brow to grow in. Around 6-9 months you can expect to see some realistic regrowth- in the interim makeup and micropigmentation can help. 

Depending on the depth of the burn, the area may recover.  If it's been only three months you want to wait not only to see if hair will start to grow back but to allow the tissue to recover and the scar to mature.  If there is no growth at 6-12 months after injury you can pursue an eyebrow transplant.  Your surgeon will inspect the area and see if it's healthy enough for a transplant or if you need to wait longer. 

A burst eyebrow that will not grow back after 8 months probably will not grow back again. An eyebrow can solve the problem.  Eyebrow transplants using scalp hair requires a special skill in identifying the ‘bend’; that is natura in each transplanted hair. So when put in by the surgical team, the natural curve of the hair must be placed to take advantage of it as it is placed.  It also must be put in laying them flat consistent with your natural eyebrow hairs
. Too many surgeons without considerable experience can’t seem to get either of these techniques right and if they don’t the eyebrow stick out rather than lay flat as they should.