Preheat the oven to 350 degrees Fahrenheit. Prepare a baking sheet with parchment paper and set aside.
In a large mixing bowl, add the browned butter. Whisk in the white and brown sugars, mixing until combined.
Crack in both eggs, along with adding the vanilla extract. Whisk together until smooth.
Pour in all the dry ingredients: flour, baking soda, baking powder, salt, cinnamon, and nutmeg. Use a rubber spatula to fold the batter together until a thick dough forms. It might not fully form by folding, so when it becomes large crumbles, use clean hands to press the mixture together until it combines into a dough.
Use a 2-inch scooper to make balls of the cinnamon dough and transfer to the prepared baking sheet. Place each dough ball about 2 inches apart from each other. Then use your fingers to create a well in the middle of each dough ball, also pressing the cookies down slightly.
Bake in the oven for 12 minutes or until the cookies have spread and crinkled at the edges. The edge of the cookies will also be slightly golden.

Peach Filling:

While the cookies bake, strain the canned peaches in a strainer, rinsing off the thick peach sauce surrounding them. Continue doing this until they are clean. Transfer the peaches to a saucepan.
In the saucepan, add all the sugars, wet ingredients, and dry ingredients at once: white sugar, brown sugar, nutmeg, cinnamon, lemon juice, water, vanilla, and cornstarch. Place over medium heat on the stove and stir together until combined.
As the peach filling heats up, it will become thicker. Bring this mixture to a boil, stirring every minute or so to prevent sticking and combine the ingredients evenly. As the mixture becomes hot and the peaches soften even more, use the utensil you are stirring with to break the peaches into smaller chunks.
Once the mixture has boiled and thickened, remove from the heat to cool.
Assembling Peach Cobbler Cookies: Once the cookies are done, remove from the oven and use the back of a spoon to press the cookies down again where you made wells before baking, to make sure there is space for the peach filling. Place peach filling into the center of each cookie to fill the well’s size.
Quickly whip up the glaze in a small bowl. Add the powdered sugar, cinnamon, vanilla, and milk all at once. Whisk until combined, and then rapidly to remove any clumps.
Drizzle this cinnamon glaze on top of the peach cobbler cookies with a spoon or by dragging the whisk across the top of the cookies diagonally. Serve these cookies and enjoy! These are best topped with a scoop of vanilla ice cream and eaten while still warm.