Definitely don't have to take anything apart. If you can't use a smart plug on the outlet, A zigbee temperature sensor mounted above or near the range would do the trick. There's also methane/gas detectors.
If you've never used Home Assistant, it's open source and completely local, and super easy to get going. You can automate nearly everything, so getting phone notifications or persistent temp reminders would be a breeze. Let me know if you have questions about that part. I'm by no means an expert, but I've been slowly automating everything in my house and it's been great.