All single multi engine aircraft can fly perfectly fine with an engine loss. You have to be more careful and follow certain procedures but other than that the plane is going to fly. Airline pilots go through extensive training and have to go through re-current training to be essentially re-certified on the aircraft usually every six months or so. Part of that training is time in the simulator going through emergency procedures just like this.
Ever flown on a twin engine plane over the ocean? Those have an ETOPS (Extended range Twin Operation Performance Standards or better known as Engines Turn Or People Swim) rating that gives the number of minutes that it can fly from a suitable airfield in the event that it would lose an engine. For instance the 787 with the Rolls Royce Trent 1000 engines have an ETOPS (as of earlier this week) of 130 minutes. So they can fly up to just over two hours from a suitable airfield and still be able to return on one engine.
Often the vertical tail is sized for engine failure at the most critical time, usually at takeoff rotation. The housefly flies in circles because there's no tail to straighten it out. The airplane will counter the asymmetric thrust with a little rudder and fly just fine.