Didn't want to hijack the other milk prompt in case this wasn't their thing, but how about if Alexander Pierce gave the Soldier enemas of warm milk? Pierce pets and makes soothing noises as the Soldier cries and struggles, praising him when he takes it all. Maybe he drugs the milk with sedatives to make the Soldier more pliant and sleepy, or with aphrodisiacs to make him beg to come, or with high-cramping laxatives just to watch him writhe and spasm in enormous distress.

+100000000000 if Pierce ultimately trains the Soldier to be incapable of feeling arousal while empty and has to beg to be filled with milk.