I feel like the other side of that central plate has your answer. Pop the central screw and see what is on the other side of that discoloration.
Looks the same on the other side, the pattern on the unburnt metal looks like galvanized steel, did the zinc burn off?
That looks like it, maybe use vinegar to dissolve the zinc?
That just looks like discoloration that sheet metal gets with repeated heating. I would investigate the controls, there are contacts that open/close as a bimetallic strip heats up and cools off (basic thermostat). if the contact are arcing or stuck it could over heat, also since you said smoke was coming from the knob. . .
I've been cooking some slow recipies that need a lot of heat (a lot of boiling water and slow stewing) under the kitchen fan, and the smell has greatly dissipated, it looks like the smoke was not a really big problem.
Do-It-Yourself, Repairs and Fixes
Share tips and tricks to keep people from throwing out that broken item. Repair before replace!