Not so sure the Track shammy is the same as the shammies in the video. The Track shammy I received with the ball has a rubber side with little knobs on one side. Not sure it can be cleaned the same way
Charlest would know, hope he sees this and posts an answer.
I've seen the track one; in fact, I bought one for my teammate who wanted that exact same idea.
The leather side is what needs to be cleaned; the rubber dot side is used to hold the ball by gripping your hand, so it doesn't slip. So it has only one side that is used to absorb the oil.
I'd suggest using the same routine (it's hand cleaning anyway), just clean the one dirty side.