I must agree with everyone that that looks like a very severe burn. Just because he's "been fine for 10 years" doesn't mean that's not the cause. Normal UTHs are VERY dangerous normally without a thermostat, and if they short out in any way they cause temps to spike even worse. I didn't read ALL the advice but I'll give my piece.
DON'T use Neosporin.
Get a plug-in lamp dimmer from a hardware store until you get a thermostat, but you NEED a thermostat.
Keep the paper towels for substrate
I honestly think that poor thing needs antibiotics. That burn looks like it's getting infected if it isn't already.
The Silvadene that I saw mentioned would be ideal, and I also think Betadine would help.
I'd also keep handling to a MINIMUM. Added stress won't help him heal.
Keep misting to a minimum
I think the biggest thing right now is getting a thermostat, and getting antibiotics. Those are my two main concerns. Good luck!