A student recently told me a story when we were discussing the dos and don'ts of treating burns. While cooking for a dinner party, they burned their hand and a guest ran to put lavender essential oil on their burn. Although skeptical, they were surprised that the burn felt better. I grew up in the era when putting butter on burns was the main treatment. (Please never do this, it is very painful and does not aid in healing). Needless to say I was a little skeptical. I enjoy essential oils for the smell so I had to do some research to live up to the CPR/Pool nerd title I have given myself.
I came across two articles I believe are from credible sources. First, the National Institute for Health (NIH) has an article on their website that describes experimental use of lavender oil for the treatment of burn wounds. The article is a little gory and not for animal lovers since they experimented on rats. However, the results recommend the use of lavender oil because it does three things: rapidly replaces collagen type I and III, increases fibroblasts (which synthesize collagen), and induces TGF-β which aids in wound contraction. This means that burns may heal faster with topical lavender oil application. You can read this article here: here:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4880962/
Another article, this one from the Mayo News in Ireland suggests cooling with cool running water (the gold standard for stopping the burning) then spritzing the burn with dilute peppermint oil added to water or applying a mixture of Aloe Vera and peppermint. This cools the burn further without dropping body heat (kind of like the cooling sensation of York Peppermint Patties :) ). To aid in the disinfection and healing, use lavender water spritz or lavender Aloe Vera. You can read this article here: https://www.mayonews.ie/component/content/article?id=7130:health-treating-burns-with-essential-oils
I don't think that either the American Red Cross or American Heart Association will add lavender or peppermint to their advice on burns. From the articles, the lavender really helps with healing and not with immediate treatment, which is the goal of first aid training. Your dermatologist however, might consider recommending calming lavender as a complimentary alternative medicine (CAM).