This is a typical effect of water loss.

- Temp goes up when the water level is dropping.
- Then down if the pressure is down to zero.
- The sensor is on top of the head and water vapor without pressure has a max temp of 100C.

Letting it idle without coolant, even for e few minutes, has certainly destroyed the head gasket and probably warped the head.

Switching off the engine when the temp rises suddenly is the only way to prevent further damage.

Sometimes you don't have a choice, I had a hose failing on the German autobahn, better damaging the head of the car than my own.

The head gasket has to be replaced and the head checked.

Depending on the state of the car and the mileage of the engine it may be more economical to replace the engine with a good used one.

Rob