I would likely try the methods of car headlight repair:
- fine sandpaper (e.g. 600)
- followed by ultra fine sandpaper (e.g. 1000)
- followed by polishing sandpaper (e.g. 2000 or even more)
- followed by polishing paste (e.g. cerium oxide) on a felt pad
However, since I see that the sink has a glossy surface... I would be deterred by that. The method I mention may reach a layer which isn't burnt, but will wear off glossy finish and there's no certainty of it returning in the same tone after polishing is done.
What paint to use - sorry, no idea.