I had this very same problem with my gorgeous stainless steel pots and pans.
and I went searching on the internet for an answer to my dilemma.
For tough baked-on stains, you want to use Easy-Off Oven Cleaner and Brillo Pads.

Image By: Andrea Jean Cleaning Via YouTube
You will spray the bottom of the pan with Easy-Off and let it sit for 2 minutes.
Then you will scrub it with a Brillo Pad and rinse.

