Slight thread drift for a moment but still has something to do with throttles.
The Boeing 737 FCTM under the sub-heading Manual Flight, states the PM should make AFDS mode selections at the request of the PF.
Sub-heading Automatic Flight, states when the autopilot is in use, the PF makes the AFDS mode selections. The PM may select new altitudes but must ensure the PF is aware of any changes.
The problem arises when the aircraft is flying with one engine inoperative with the autopilot engaged. In normal two engine flight the rudder is rarely used. In asymmetric flight, rudder is used especially where thrust changes mean rudder adjustment to correct for changing yaw. In asymmetric flight the autothrottle is switched off as per QRH.
So what we have in the one engine inoperative situation, with autopilot engaged is actually a combination of manual and autoflight. That is, manual throttle and manual rudder plus autopilot pitch and roll.
Because of the half automatics and half manual tasks associated with one engine inoperative, it then becomes an unwieldy situation where the PF makes changes to the MCP involving airspeed and heading and even altitude changes, while at the same time having one hand on the live throttle and maybe also making rudder trim adjustments. The one arm paper hangar comes to mind.
Clearly Boeing should amplify its advice on MCP changes by the PF when on one engine and the automatic pilot is engaged. . It would be sensible to have the PM make MCP changes at the request of the PF during asymmetric flight. This leaves the PF with the task of simply operating the live engine thrust and adjusting rudder pressure changes backed up by manual rudder trim if required.
While this becomes a matter of common sense, the fact remains there are pedants in the simulator who arc up if the Boeing FCTM advice is not followed to the letter