Hi there and welcome!
My guess is that the high temp safety switch has a loose connection or the connector is corroding away.
This causes resistance which then causes a major heat up of the metal/wire/connector and the switch at that point.
The switch is now heating up like crazy, much hotter than your smoker and it is reading it's own super hot temp instead of the smoker temp and is cutting power off to the heating element.
Once this safety switch reads a temp over it's limit it stays open for a while thereby not allowing power to the heating element until things cool down. The problem at this point is that the switch itself has to cool down, not the smoker and the switch cooling off can take a lot longer.
You can confirm this by accessing the safety switch from the back of the smoker and checking things out.
You would see burning/melting signs or corroded connectors or something odd. You can always cut the wires off that switch and connect the ends and see if that solves the issue as well and if so you know the safety switch was failing you.
WARNING: if you do this, know you have removed the safety mechanism for overheating. You are accepting responsibility for doing this for troubleshooting, or to get through an emergency cook, or you feel you can handle going forward without a safety switch (or until you can replace it).
If you are very luck you have a panel already there midway up the smoker usually on the right side or in the middle.
Most MES units do not have a panel here so you have to cut one which is annoying BUT it is super handy as these cheap plastic switches are a common failure point so having a panel cut in allows you to access the switch and bypass or fix so you can keep going without too much of a hassle vs ruining a whole smoke :D
This post here will show you where this safety temp limit switch is located on the inside of the smoker as well as on the back:
The MES safety rollout limit switch is a common point of failure so I am making this post to have pictures of it that I can easily find instead of digging up new ones everytime we help someone with this issue. Also adding tags so it's easy to search in SMF and hopefully via Google as well...
www.smokingmeatforums.com
This is most likely your issue with what you describe. I hope this helps and let us know what you find :D