I have a charcoal grill (Big Green Egg) that has a ceramic insert that can be used as a heat shield or as a pizza stone. My stone is black from drippings and smoke. I'd like to cook some pizza on it, but I'm not sure what I should do to clean it.
Perfect. Sounds like your pizza stone is nicely seasoned. Scrub it with your stiffest brush, rinse with water, no soap, done. If you're paranoid about germs, cook it before cooking on it. Throw it in the oven at a few hundred degrees, for 15 minutes or so.
Ideally, you're supposed to heat the stone (thus sterilizing it) before slapping the pizza upon it anyway (although that requires a pizza peel).
I concur with all of the people above that the black burned on parts aren't at all a problem.
However if you want that buff color back, I got it back on one of mine accidentally. I had a pizza stone that basically lived in my electric oven. During some holiday or another I stuffed it under the lower burner of the oven, and forgot it.
Later I ran the electric oven self cleaning cycle. When I opened the oven I was shocked to see a big buff colored disk in the bottom of the oven! It had been black for so long I hadn't seen it when I pulled out the thermometer before cleaning the cycle and had forgotten it was down there.
I don't know that it won't break your Big Green Egg Plate Setter. I use the aforementioned pizza stone on top of mine to keep the BGE part cleaner.
I have used this technique a few times to clean up pizza stones others thought were "ruined' or "finally seasoned." But with an Egg part I suppose I would worry that it might break, and I would have to replace it. Pizza stones are much cheaper than Big Green Egg parts.
Generally, I prefer putting it in an oven and using the self-cleaning function when there is a fewer amount of oil or baked staple foods stuck on the surface of a pizza stone. You can also use a pizza scrubber brush. Wash with baking soda.