Two words:

Barkeeper's Friend

Seriously, that stuff can do wonders. You have a pretty bad burn there, but I definitely won't be surprised if Barkeeper's Friend brings it back to mint condition. We have a complete set of All-Clad too (there's nothing like it), and when we aren't feeling lazy we'll use a little of that stuff on a sponge to get it looking nice again.

ps- I linked to some spray bottles because I was excited to see it in that form. Most of the time it's either in a powder or a squeeze bottle. I'm definitely going to try the spray next time, just for convenience...