Perhaps i'll try and address the OP:
Are you enquiring as to our actions if we have some sort of engine problem that directs us to run the engine at idle or close to idle (in your example, in an OVHT or maybe a surge or stall) yet not shut it down?
We have had this in the sim recently. Obviously, to shut down a working engine, even if only generating idle thrust, is silly. It is still providing viable electrics, hydraulics, and pneumatics. In this situation we would run it at idle and then go to the One Engine INOP landing checklist.
Remember that the QRH is just a checklist - it cannot cover every situation and this is one of them. Just because you aren't specifically directed to an appropriate checklist, doesn't stop you from using your brain and choosing the best course of action.