Combine all of the ingredients, besides the lemon zest, in a large saucepan.
Allow the mixture to come to a simmer (it begins to release bubbles around the edges), and then reduce the heat slightly to a low simmer for 15 to 20 minutes. Stir occasionally with a wooden spoon or spatula. If you notice the mixture sticking to the bottom of the pan, reduce the heat further.
During the last few minutes of the process, add the lemon zest and stir thoroughly. Remove from the heat.
Allow the mixture to cool slightly, and then using an immersion blender blend the mixture until smooth. If you don't have an immersion blender, wait until the mixture is cool and then transfer to a traditional blender or food processor to blend.
When the peach butter is cool, transfer to storage jars or airtight containers. The peach butter will last in the refrigerator for up to 7 days. You can freeze the peach butter for up to three months in an airtight container or in a glass jar (be careful to leave a bit of extra room when freezing it to allow for expansion). To use, remove from the freezer and allow to defrost in the refrigerator overnight.