Instructor pilot reported Engine Failure during stall recovery training. The Instructor Pilot immediately assumed the controls and the flight crew asked ATC for vectors and began to divert. The engine recovered without any input from the flight crew; so the flight crew continued to divert and make a precautionary landing.

Narrative

I was conducting check ride preparation to the south. The Engine died during a power on stall recovery during pitch down after the break. I immediately called for my controls from the Student. I pitched to best glide and tried to start the Engine a couple of times from memory because the Student had the checklist on his side. After I could not start it I asked the Student to grab the checklist and go through the restart process while I continued flying toward a point of landing and contacted ATC.I contacted ATC and was given vectors to ZZZ airport. I quickly realized I would not make it and decided to turn toward my best place to land. As soon as I turned towards my best place to land the Engine came back on without input from me or my Student. I regained thrust but not total thrust and landed at ZZZ. I called ATC when on the ground and asked if there was any paperwork or numbers to call and they told me there was none. I also asked my Boss if there was a process I need to go through with the School and there was none for me to do.