Try liquid cleaner for ceramic top electric stoves, I don't think that this is abrasive and if it will clean burnt on stuff from a stove top it should also clean the enamel too. It also tends to leave a shiny finish when you're done.
Eddie
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Our NYC apartment has a very old Wel-Bilt 20" gas range. The drip insert was so encrusted
with many decades of burnt, cement like food build up, and rust, that recently i bought a brash wire wheel which i installed on my power drill and went to town.
OMG it was hard work. So hard i left half the job for next time i'm feel like doing something sorta insane.
It worked though. By the looks of things the brass wheel will be toast after i finish the job.
I'll post pictures later.
So, no i do not know about the whole "without scratching" question and probably should have post my tale in it's own thread...that said the scratching of the black surface was minimal. Must be porcelain.
Like Eugene said treat it like glass because that's what it is. Porcelain enamel aka Vitreous enamel is a form of glass. Powdered glass is melted and fused to the metal under high heat.
Ammonia is the active ingredient in oven cleaner. Another trick is to take the drip trays and put them in a bag with ammonia. Seal it up and wait a few days. It will dissolve most of the grease. If the cooktop is greasy, pour ammonia on the top, use Saran Wrap to seal it.
Last trick is to stick the parts in an oven that has the self clean feature. If the chemicals won't work, the heat will turn it all to ash.
I've been cleaning greasy stoves for 15 years. I've likely cleaned thousands at this point