Further, there is no one single dimension you can rank the 'best' professor on. The two main dimensions would be Research and Teaching. A third might be Public

Further, there is no one single dimension you can rank the 'best' professor on. The two main dimensions would be Research and Teaching. A third might be Public Engagement. But there's not much correlation between these. When I was doing my PhD in Physics I've seen professors doing brilliant research but horrible at teaching or explaining concepts in an understandable way. Richard Feynmann was a rare exception excelling in all three dimensions above.