I would likely try the methods of car headlight repair:

• fine sandpaper (e.g. 600)
• followed by ultra fine sandpaper (e.g. 900)
• followed by polishing sandpaper (e.g. 1200)
• followed by polishing paste (cerium oxide)

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 may wear off glossy finish.

What paint to use - sorry, no idea.