Preheat your oven to 350°F. Line a baking sheet with parchment paper.
In a medium bowl, whisk together the flour, baking soda, salt, cinnamon, and nutmeg. Set aside.
In a large mixing bowl, cream together the softened butter, brown sugar, and granulated sugar until light and fluffy, about 2-3 minutes.
Add the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.
Gradually add the dry ingredients to the wet mixture, mixing until just combined.
Fold in the diced peaches, rolled oats, and chopped pecans if using.
Drop rounded tablespoons of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake for 12-15 minutes, or until the edges are golden brown. Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.