Preheat your oven to 350°F (175°C). Line a baking sheet with parchment paper or a silicone baking mat and set aside.
In a large mixing bowl, using a hand mixer or a stand mixer fitted with the paddle attachment, cream together the room temperature butter, brown sugar, and granulated sugar until light and fluffy, about 2-3 minutes.
Scrape down the sides and bottom of the bowl with a rubber spatula. Add the room temperature egg to the butter and sugar mixture. Beat on medium-low speed for 30 seconds until fully incorporated and the mixture is smooth.
In a separate bowl, whisk together the all-purpose flour, cornstarch, rolled oats, cinnamon, nutmeg, baking powder, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing on low speed until just combined. Be careful not to overmix. Scrape down the sides and bottom to ensure the mixture is evenly combined.
Use a 3-tablespoon-sized cookie scoop to portion out the dough. Push two scoops of dough together to form larger cookies. This will make seven large cookies. In the palm of your hand, roll them into balls.
Place 3-4 dough balls on the prepared baking sheet, spacing them about 3 inches apart.
Bake in the preheated oven for 14-15 minutes, or until the edges are golden brown but the centers are still slightly soft. The cookies will continue to set as they cool.
When the cookies are done baking, remove them from the oven. While they are still warm, use a flat spatula or the bottom of a cup to gently press the cookies down to about ½ inch thick.
Allow the cookies to cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.
FOR THE PEACH FILLING
Peel, core, and dice the peaches into small ½-inch cubes, ensuring the skins are removed. You can also use frozen peaches.
Add the diced peaches to a medium saucepan over medium heat. Add water, brown sugar, cinnamon, and salt,
Cook the mixture until the peaches until the are soft and tender. Fresh peaches will take the longer to cook.
In a small bowl, combine the cornstarch with cold water and whisk to create a slurry.
Pour the slurry into the hot peach mixture and cook for 3-4 minutes until the filling thickens and turns clear.
Remove the mixture from the heat and stir in the lemon juice. Mix well to combine.
FOR THE CRUMBLE TOPPING
Preheat your oven to 400 degrees F. Line a baking sheet with parchment paper and set aside.
In a medium bowl, the brown sugar, flour, butter, cinnamon and salt. Mix by hand with a wooden spoon until the mixture resembles clumps of sand.
Spread the crumble over the parchment paper and bake for 8 to 10 minutes, stirring halfway through. The crumble should be golden brown. Let it cool completely and then crumble the mixture into smaller pieces if the pieces are too big.
FOR THE CREAM CHEESE FROSTING
In the bowl of an electric mixer, combine the butter and cream cheese. Cream on medium speed for about two minutes. Scrape down the sides and bottom of the bowl. Beat again for 2 more minutes.
With the mixer on low speed, add the powdered sugar, followed by the vanilla, ground cinnamon and salt. Beat on low just until mixed. Then turn the mixer to medium high speed and beat the mixture for 2-3 minutes.
ASSEMBLY
Once the cookies are ready to be assembled.
Start with placing the cinnamon cream cheese frosting in a piping bag or a zip lock bag cutting a medium size hole. Set aside.
Take a cookie and place 1-2 tablespoon size of the peach filling on top of the cookie. Evenly spread it across the top.
With the piping bag, frost a swirl right on top of the filling.