PA-23 flight instructor reported left throttle stuck at full power. Instructor shut down engine and returned to departure airport.

Narrative

Myself and one student departed for a training flight in the Apache. As we were climbing through ~3;400 ft. MSL I pulled the left throttle to simulate an engine failure with my student. Immediately upon pulling the throttle I noticed no friction on the knob; as well as no change in manifold pressure or RPM. I determined that the throttle cable had most likely snapped and our left engine was stuck at a climb setting. After evaluating the situation I instructed my student to level off the climb and make a turn towards the airport. I contacted tower frequency to advise that we would need to orbit over the field until we had a plan of action. Despite being above the Class Delta airspace I wanted to establish communication with tower to plan for a landing. I tuned into UNICOM to discuss the matter with a pilot on the ground; who was the one who instructed me towards my multi and MEI; for an educated opinion on the matter. After a brief discussion on the radio I elected to initiate a descent with both throttles at idle (left throttle still stuck at full). I told tower I was having mechanical issues and advised that I would need to make an urgent full stop. I elected not to [request priority handling]. As I was on an extended left downwind for Runway XX at approximately 2;700 ft. I extended the landing gear. As I turned for about a 3-4 mile final I determined that I was too high and too fast for my final. I then shut down and feathered the left engine and continued my approach single engine to land on runway XX. I successfully touched down around the intersection of runway XX and runway YY and rolled out to a complete stop on the runway and then proceeded to single engine taxi off of the runway onto taxiway 1. From taxiway 1 the aircraft was towed to maintenance.