Hi there and welcome Mark!
Yeah I see that burnt smooth black wire at the lower circuit board. I couldn't tell if the little plastic box component it plugs into was swollen up and showing burn signs either. If the little plastic box component seems fine then it may be that you can get away with changing the wire connector on that black wire and hooking back up and see if it works.
My hunch is that if the black wire is burnt up then there was looseness with the wire and the connector, or the connector and the black plastic box component. So much so it causes resistance and that causes major heating up. Enough to melt plastic insulation or that little black plastic box component.
If you switch the connector and it all works then you are good to go!
If the circuit board or that black box component on the board itself are burnt up then you will have one hell of a time finding a replacement part. If you are a circuit board hardware kind of guy you might be able to replace that component but I've only seen 1-2 people savvy enough to do so (I'm not one of them lol).
The other option (better option actually) is to do the simple rewire and buy a ready to go PID controller and turn that golf cart of an MES into a Ferrari :D
The PID controller runs like $160 BUT it really is a whole different smoker and one that performs like similar electric smokers with PID controllers but they cost over $1,000!!!
So for $160 and no more effort than you plan to apply already, you can have a smoker that hits and holds dead on with temps and you can keep running through anything short of it being hit by a car lol.
Here is the simple rewire guide for using a PID controller. Simple = cut the ends off 4 wires in your picture and then wire nut to make 2 whole wires. At that point the MES plug will feed power to the heating element keeping the high temp safety switch in the mix while bypassing ALL other stock MES electronics that tend to fail.
Lately I've seen a number of posts about guys wanting to rewire their MES. I had wanted to post a quick and simple guide for those out there that are curious or those that simply need to rewire but need a little assistance. Well here goes. Disclaimer: When messing with electrical equipment...
www.smokingmeatforums.com
I hope this info helps and ask all questions you have :D