Controlling 800 W ! I will use a relay rated to switch at least 15 A - 120 V ac. And work it like an electric thermostat, On and off, on, and off... A sensor - thermocouple ? That might work... I don't know where to find one of them.
It might also have a "Zero Crossing" switch (detects when the AC sine wave is at 0V to turn off)
That is a dimmer switch technique. Controlling a heather just like a dimmer is not to practical in my opinion. A simple high current rated relay will work just fine. I think the stove element work on that principle. I can hear clic and clic again when it is too hot. Hum, I wonder what type of sensor the stove use... maybe the same sensor the stove use "migh" work ... 