It may, in fact, be a lot like a scab, and in the days ahead may flake off slowly like a scab. You can put some neosporin on the gauze if you want, but at some point it will be a good idea to let the burn air-dry. If it has the texture of a scab (dry, somewhat crusty) then the neosporin is less important than just keeping it clean and protected.
Neosporin does not normally sting, it's basically vaseline with an antibiotic in it. I've used it on cuts & so forth, and its never stung me.
Make sure you regularly clean it gently; use some mild soap and cool water, and gently wipe with a washcloth. Rinse off the soap with cool water, then dry it by blotting (gently press a towel on it, don't rub).
EDIT: A great deal depends on how deep the burn is, which is determined by how hot the 'branding iron' was, and how long you had it against your skin. Apparently it was not hot enough to give you 3rd degree burns, which would have caused your skin to be black & crispy right afterward.
The damaged tissue will be slowly moved to the surface and shed. I think I've heard that normally, a human's skin is totally replaced every 4 weeks anyway. If I'm remembering right, that should, logically, be the maximum time it should take to shed all the damaged layers of skin.
Until then, you just need to prevent bacteria from growing in the area. Because you have dead cells in there, the badly burnt areas can no longer be protected by your immune system. That's why we keep it as clean & as sterile as possible.