Already been done with EU funding
Several years ago I was working on a project involving the microwave heating of plastics. I had access to an industrial grade microwave oven which not only had completely variable power output but also a variable speed turntable. There were chokes in the side and top of the oven to allow viewing of the contents without microwaves leaking out. A military spec IR camera and image analysis software completed the experimental set up. The opportunity to misuse this equipment to heat one's lunch was recorded for posterity: https://www.youtube.com/watch?v=caN-PFSxzG8
Thermocouples and thermistors (even shielded ones) are not very good at measuring the temperature in a microwave field as they act as antennas and distort the results. As an alternative to thermal imaging we used fibre optic temperature sensors which work by measuring the decay time of a fluorescent agent when pumped with a flash gun.
As has already been said, most modern domestic microwaves heat on a duty cycle by simply turning the filament in the magnetron on and off. It is a fairly simple hack to convert one into a continuously variable power output one using a variac.
We had some pretty exciting times when our specimens caught fire inside the oven as hydrocarbon flames and smoke (containing finely divided carbon) couple very strongly with microwaves and things rapidly escalate.