Preheat your oven to 350°F and line a baking sheet with parchment paper.
Cream the butter, brown sugar, and white sugar together in a bowl until light and fluffy.
Add the egg, vanilla extract, and almond extract. Mix until smooth and well combined.
In a separate bowl, whisk together the flour, cinnamon, cream of tartar, baking soda, salt, and nutmeg (if using).
Slowly add the dry mixture to the wet mixture, stirring until a soft dough forms.
Fold in the chopped fresh peaches gently.
Scoop dough onto the lined baking sheet, spacing them 2 inches apart. Bake for 10–12 minutes until edges are lightly golden.
Cool on the sheet for 5 minutes before transferring to a wire rack.
Notes
Store in an airtight container at room temperature for several days or freeze for longer freshness. You can also freeze the dough before baking for easy cookies on demand.