I have the same Cuisinart set as you do and have had far too many similar disasters as you, from the kids, etc. It IS a stainless steel pot (set) with NO non-stick, no coating of any sort, so you CAN use an oven cleaner on it if that is something you wish to try.
I have worked off as much as I could using BKF, scouring pad, etc., and when nothing further could be done, that is when the oven cleaner is called in! Get the pan hot by boiling water in it first, toss out the boiling water, then immediately spray with the oven cleaner. Have the lid to the pot ready, to cover it, let set per the oven cleaner directions.
If you do not want to use the oven cleaner, I have also gone to the point of boiling the water and then adding 1/4 cup of dishwasher soap....the crystals, not liquid, as for some reason the liquid dishwasher soap just doesn't work like the dry does. Add the dry soap to the boiling water and let simmer for approx. 30 minutes. Remove from heat and check to see if the black particles have removed themselves. If not, try to scrape them off with the scouring pad. You may have to return the pot and soap mixture to a boil and repeat. It does work.