Preheat the oven to 350°F (175°C). Line baking sheets with parchment paper.

Make the crumble: In a small bowl, stir together flour, brown sugar, and cinnamon. Cut in cold, cubed butter with a pastry cutter or forks until coarse crumbs form. Refrigerate while you make the dough.

Whisk dry ingredients: In a medium bowl, whisk all-purpose flour, baking powder, baking soda, salt, cinnamon, and nutmeg.

Cream butter and sugars: In a large bowl, beat butter with granulated sugar and brown sugar until light and fluffy, 2–3 minutes.

Add eggs and vanilla: Beat in eggs one at a time, then mix in vanilla until smooth.

Combine: Add dry ingredients to the wet in two additions, mixing on low just until no dry streaks remain.

Prepare peaches: Pat diced peaches very dry with paper towels. If very juicy, toss with 1 teaspoon flour.

Fold in peaches gently with a spatula until evenly distributed without crushing the fruit.

Portion: Scoop tablespoon-sized mounds of dough onto prepared sheets, spacing about 2 inches apart.

Top: Sprinkle each mound generously with the chilled crumble, pressing very lightly so it adheres.

Bake 12–14 minutes, until edges are lightly golden and centers look just set. Cool on the sheet 5 minutes, then transfer to a wire rack to cool completely.

Optional glaze: Whisk powdered sugar with enough milk to make a drizzle. Lightly glaze cooled cookies and let set before serving.