Pulling the mixture, in most training aircraft since about 1960, does not shut down the engine at all.
I am getting a bit slow in my dotage and for that reason could you run that by me again? if you cut the mixture the EGT and CHT drop sharply (CHT more slowly of course). The aircraft yaws sharply depending on the power being applied to the second engine. The `dead engine propeller windmills at a rotation rate caused by the indicated airspeed at the time and for all intents and purposes the engine is dead. And now you are staying the engine is operating?
Try as I may I cannot understand your statement that the mixture cut does not shut down the engine.