@seanv, With a Magic Eraser to clean off the oxidation first after cleaning up the bowl in the sink. I usually wash it out and clean it with many pipe cleaners first, then I dried out the bowl, set it to the side and used the Magic Eraser and as hot a water as I can stand, and then just scrub it until all the brown is gone. Then yesterday, I sat down with some hobby sanding sheets that range from 150 to 600 grit and wet sanded all the divots out. Took longer than I thought it might. After that, I sat down inside with my 3,200 - 12,000 grit micro mesh sanding pads and sanded/polished my way to a beautiful mirror black. Once I was satisfied with the outcome, I spread a nice layer of my butchers block conditioner over the whole pipe again and let it rest.
Yeah, it really did clean up beautifully! Even though those depressions were deeper than I originally thought. The 220 grit sandpaper just wasn't doing the job fast enough in my opinion, so I had to reach for the 150 grit sheet.