Combine sliced peaches, sugar, and water in a large saucepan.
Bring to a boil over medium-high heat, then reduce heat to low and simmer, stirring occasionally, until peaches are very soft, about 20-30 minutes.
Mash the peaches with a potato masher or immersion blender until smooth.
Stir in cinnamon, nutmeg, and salt.
Continue to simmer, stirring frequently, until the mixture has thickened to your desired consistency, about 1-2 hours. The peach butter should be spreadable but not runny.
Remove from heat and let cool completely.
Transfer to jars and store in the refrigerator.
Notes
For a smoother texture, process the cooked peaches in a food processor before adding the spices.