I have used baking soda made into a paste and applied to the inside of my oven to clean it with great success. The down side to this is that you leave it on for a few days and may have to reapply to some stubborn areas. But maybe you could try it with one at a time and not use that burner.
I would also think that anyone who manufactured this range would have suggestions on how to clean it, or what they suggest (since they have suggested that you do not use steel wool). If it isn't listed in the user guide, see if there website has suggested cleaning products. Damned fancy-smancy appliances anyways!!!!!!!!
The dumbest suggestion I have, for after you get them clean, is to wash them all the time to keep this from getting out of control. 
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