Hello,
I agree with the doctor because it looks like a superficial burn. Most superficial burns heal within about two weeks and don't usually leave a scarf. For these kind of burns; you can apply antibiotic cream such as silver sulfadiazine (Silvadene).It will prevent infection and help in fast healing.
One medicine which has been advised by many dermatologists for scar diminishing is contractubex gel/cream. You can seek your dermatologist/surgeon’s opinion before using this cream. Otherwise you can go in for glycolic peals, laser treatments and pressure dressings. Severe scarrings may need skin grafting by a plastic surgeon.
I hope it helps. Take care and regards.
If you're doctor says its a first degree burn, then I'd tend to believe him since its easy to diagnose this skin condition.
You can help heal the burn and lessen the formation of scarring by applying pure aloe vera and pure vitamin E to the effected areas on an ongoing basis. Keep the skin covered with gauze so that the skin doesn't dry out. Keep using fresh gauze or bandages until you can see that a new layer of skin has generated. Keep applying the aloe vera and vitamin E even after you've stopped using the gauze. Hope you have relief within the next couple of months.