Line two baking sheets with parchment paper and set aside.

In a medium bowl, whisk together the all-purpose flour, cake flour, baking powder, baking soda, and salt. Set aside.

In a large mixing bowl, using a stand mixer or hand mixer, cream the cold butter on medium speed for 30–45 seconds. Add the brown sugar and granulated sugar and continue mixing until the mixture becomes pale and slightly creamy.

Gradually add the dry ingredients to the butter mixture, mixing on low speed. The dough will appear crumbly at this stage. Add the beaten eggs and mix until a thick dough forms. Refrigerate the dough while preparing the crumb topping.

To prepare the crumb topping, combine the flour, brown sugar, cinnamon, and melted butter in a small bowl. Mix with a fork until pea-sized crumbs form.

Remove the dough from the refrigerator and fold in approximately ¾ of the crumb topping, mixing gently. The mixture may remain slightly uneven.

Portion the dough into large balls weighing approximately 6 ounces each (about ¾ cup). Form 5–7 cookies. For each cookie, remove a small portion of dough from the top, press a shallow well into the center, and fill with chopped peaches and additional crumbs. Cover the filling with the reserved dough and reshape into a sealed ball. Sprinkle extra crumbs on top if desired.

Place the filled dough balls on the prepared baking sheets and transfer to the freezer for 90 minutes.

When the cookies are nearly finished chilling, preheat the oven to 375°F (190°C). Place one empty baking sheet upside down on the center oven rack. Position the baking sheet with the cookie dough directly on top of the inverted pan.

Bake for 24–26 minutes, or until the edges are lightly golden brown. Remove from the oven and allow the cookies to cool on the baking sheet before serving.