Forewarned is forearmed. Thanks. 
I don’t know if they make it anymore, but some years ago I swore by an ointment called Foille. Fantastic stuff, although I never had a burn beyond first degree to try it out on.
It contains benzocaine. Yep, it would be good stuff, when used appropriately.
Out of curiosity, I looked it up and yes, Foille Ointment is still available for purchase. I may get some just in case.