[cheshired5]

First job is to check the coolant level on a stone cold engine. It must not be above max or below minimum.

"To the brim" sounds excessive but it is normal for coolant to rise above the Max line hence the term expansion tank. If it rises too high though, there may be a flow restriction, airlock in the system or loss of pressure from the expansion tank cap for example.

Even cheap diagnostics will give you live data to view the coolant temperature behaviour so you can see if the thermostat is opening as it should.