I agree that cooling, cooling, and more cooling is the best first aid.

Immersion in cool water really works for burns to the extremities. If it hurts when you take it out, you need to put it back in. Keep it in cool water for as many hours as it takes. The results next day are near miraculous.

I don't know where the line is for very serious burns. Would the reduction in damage from immersion be worth the increased risk of infection?