Based on the question, my assumption was that the slate has some foreign material on it that requires the removal of some of the stone itself from the call, and that the problem was not just a superficial blemish. As you must know, slate is a highly porous material. That is why I said to use sandpaper if the oil is not too deep. I would expect if the slate has been exposed to oil, as described, a layer of the stone itself would have to be removed to make the slate usable again, and that may not even work if the oil has penetrated all the way through the stone, in which case a possible remedy would be to try to bake the oil out with heat, but this would probably damage the pot. Of course, for routine maintenance scotch-brite is sufficient.