This can only be done in the First Aid Level 3 minigame, with Scrap Baby, as that's the only one that gives amputation/replace parts objectives. First is making sure you have what you're looking for. That being, either a leg amputation, brain replacement, or eye replacement objective. As quickly as possible, try to get the part take off of Helpy. Then, with it still in your grip, allow Scrap Baby to approach. This is the most crucial part, is making sure the part actually hits Scrap Baby. If it misses, it won't count, and will not give you the achievement.
You will fail the Minigame by allowing her that close, but the arc for throwing pieces in Flat Mode is really short. You will not get enough time to put the mask on him after the attempt. So, you will only get one shot at it per run.