Preheat oven to 350°F (175°C) and line two baking sheets with parchment paper or silicone mats. Chop dried peaches into ¼-inch pieces; if very dry, soak in warm water for 5 minutes then pat completely dry.
In a medium bowl, whisk together flour, baking powder, baking soda, salt, 1 tsp cinnamon, and nutmeg for about 30 seconds until evenly combined. Set aside.
In a large bowl, beat softened butter, granulated sugar, and brown sugar on medium speed for 3–4 minutes until light, fluffy, and pale, scraping down the sides as needed.
Add eggs one at a time, beating well after each addition, then mix in vanilla extract for 30 seconds until the mixture is smooth and creamy with no streaks of egg.
With the mixer on low, add the flour mixture in three additions, mixing just until no flour streaks remain after each — do not overmix. The dough will be thick and slightly sticky.
Using a wooden spoon or spatula, gently fold the chopped dried peaches into the dough until evenly distributed throughout.
In a small bowl, stir together the turbinado sugar and 1 tsp cinnamon until combined. Set near your work area.
Scoop dough into 1½-tablespoon balls, roll between your palms, then dip the top of each ball into the cinnamon-sugar mixture. Place cinnamon-sugar side up on prepared baking sheets, spacing 2 inches apart.
Bake one sheet at a time on the center rack for 11–13 minutes, rotating halfway through, until edges are set and just golden — centers should still look slightly soft. They will continue to set on the hot pan.
Let cookies cool on the pan for 5 minutes, then transfer to a wire rack to cool completely. Repeat with remaining dough. Enjoy warm or at room temperature.