How to best clean this pan?

We got these All-Clad HA1 Anodized Nonstick Fry Pan Set about two years ago, and they've been great. But despite only using the smaller one for eggs, it's acquired this black buildup on the sides. I can scrape it off with a fingernail, so I don't think it's the metal or a coating coming off (probably just olive oil?). We wash with a non-scratch sponge and soap and water after letting it cool.

Is there a proper way to clean without being too invasive? I've seen suggestions of mixes of vinegar and water, but wanted to see if I got any different feedback for best ways to maintain these things and/or avoid this in the future.