Prep the peaches: Dice into small pieces, blot with paper towels, and toss with cornstarch. Set aside.
Preheat oven: Heat oven to 350°F (175°C). Line baking sheets with parchment paper.
Cream butter and sugars: In a large bowl, beat softened butter, granulated sugar, and brown sugar until fluffy (2–3 minutes).
Add egg and vanilla: Mix until just combined.
Mix dry ingredients: In another bowl, whisk flour, baking powder, baking soda, salt, cinnamon, and nutmeg.
Combine wet and dry: Gradually add dry mixture to wet ingredients. Mix until just combined.
Fold in peaches: Gently fold diced peaches into dough with a spatula.
Portion dough: Scoop about 1.5 tbsp per cookie, place 2 inches apart on baking sheets. Chill briefly if dough feels soft.
Make streusel: Mix flour, brown sugar, cinnamon, and salt. Cut in cold butter until pea-sized crumbs form.
Top cookies: Press streusel onto each cookie dough ball.
Bake: Bake for 10–12 minutes, rotating halfway, until edges are golden and centers just set.
Cool: Let cookies rest on sheet for 5 minutes, then transfer to wire racks.
Glaze (optional): Mix powdered sugar, vanilla, and milk. Drizzle over cooled cookies.