Cutting up the peaches doesn't have to be pretty, or uniform, because they're going to be pureed down anyway. Once the peaches have been added to the sugar, let them mingle for a bit while you move on to make the ice cream base.
The mixture will look light, lemony yellow once the milk has been added in. Now that you have safely navigated that part, pour everything back into the saucepan to heat through. I always pour mine through a fine mesh colander, just in case some eggs may have scrambled. No one wants scrambled eggs in their ice cream!
Now, if you have an immersion blender, use it here to puree your peaches! If you don't, go on and puree these peaches down in a blender, or a food processor. You can puree it as finely as you want; if you want some chunks of peaches, leave it as chunky as you want! I like mine with a few chunks, but its your ice cream.
This is not the correct way to cover your ice cream base. Your base will form a skin, and that will not be tasty in your ice cream. That being said, here is the proper way to cover your ice cream base. Gently press the plastic wrap down against the base, and up the sides of the bowl.
Yep, I had to stop my machine and try to flip my churner thing around. Unfortunately, a lot of the ice cream had started to set up, so I ended up just making a mess. However, I knew I needed the churner thing in there, because it is what puts air into the ice cream, which is what helps make the texture.