If I can add a bit
Don't just bring the throttle back on the suspect engine. Take it up and then back. I know we started talking about EFATO but I have seen pilots when given an engine failure at a low power setting mis-identify. If you take the throttle up from a low setting and nothing happens then that is the dead one.
As for cowl flaps with EFATO on a PA31, close both sides then open the live side once you have enough height. There is a lot of drag in them and the first priority is to get some height. The engine will last with the cowl flaps closed for a minute or so.
Also on the the PA31, if you are going to go, always select the gear up. IF the pump on the live engine is good the handle will reset to up neutral. If not, you know you may be faced with pumping the gear down and will have to allow time for that. Too late on late final to discover you had one sick pump and the gear dangling and not locked.