Preheat the oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a medium bowl, whisk together the flour, baking powder, and salt.
In a large mixing bowl, beat the softened butter and granulated sugar until light and fluffy, about 3–4 minutes.
Add the egg and vanilla extract, mixing until fully incorporated.
Gradually stir the dry ingredients into the wet ingredients until a soft dough forms.
Gently fold in the diced peaches, taking care not to overmix.
Scoop rounded tablespoons of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake the Italian Peach Summer Cookies for 18–20 minutes, or until the edges are lightly golden and the centers are set.
Let the cookies cool on the baking sheet for 5 minutes before transferring them to a wire rack.
Once completely cool, dust generously with powdered sugar before serving.