When I posted my question about cleaning charcoal dust stains off of my Eggsshelves, several folks asked me to report on what worked for me. So as requested I'm back with what worked for me. Several folks suggested Bar Keeper's Friend and I had heard good things about it before this, so this morning I picked up a can of the powdered version.
As you can see from the pictures below it did the trick nicely. At first I didn't think it was going to work, but when I swirled it around with the "abrasive" side of the Scotchbrite sponge the stain came right up. It cleaned any and all stains on the shelf.
The first thing I tried was Bar Keeper's Friend in powder form together with Scotchbrite sponges.
Here is the Eggshelf with several charcoal dust stains that wouldn't lift with anything I'd tried to date.
I wet the shelf down, sprinkled the powder on the stained area, waited a minute and then wiped it down with the Scotchbrite sponge. Until I used the abrasive side and a swirling motion it didn't appear like it was going to work.
Once I used a swirling motion with the abrasive side of the sponge all stains came up. Any dark marks you see in this picture are the shadows of distance trees in my woods.
Thanks again to everyone who made suggestions, I plan to try out some of the other products as well. But for now I have a winner.