How to clean an all-clad stainless steel pan
Brillo/S.O.S pads (the real metal ones) and Barkeepers Friend.Recently I found a liquid Barkeepers Friend that works well also.

Use the Barkeepers Friend first or just trade off.  Brillo/S.O.S last.

There has never been crud or discoloration that has survived these two products. There have never been scratches with Brillo/S.O.S on my stainless steel both inside and out All Clad.

I think that Brillo/S.O.S works to really put a shine on the interior of the pan.  Eggs do just slide around.  Not too shabby.