In a saucepan over medium heat, combine the diced peaches, granulated sugar, light brown sugar, cinnamon, nutmeg, ginger, and lemon juice. Cook for 6 minutes, stirring occasionally, until the peaches begin to soften and release their juices. Stir in the corn starch and water, cooking for another minute until the mixture thickens slightly. Remove from heat, stir in the vanilla bean paste, and transfer to a bowl to cool in the refrigerator while you prepare the cookie dough. I like to let this cool completely so it doesn't warm up the cookie filling when assembled.

In a large bowl, cream together the room-temperature butter, 1 cup granulated sugar, and 1/2 cup light brown sugar for 1 minute until light and fluffy. Add the 2 tsp vanilla extract and eggs, mixing until fully incorporated. In a separate bowl, whisk together the flour, baking soda, baking powder, salt, 1.5 tsp cinnamon, and 1/2 tsp nutmeg. Fold the dry ingredient mixture into the wet ingredients until just combined—be careful not to overmix. Chill the dough in the refrigerator for 30 minutes.

Remove the chilled dough from Step 2 and scoop it into 1-inch balls using a cookie scoop or spoon. Place the balls on a parchment-lined baking sheet and return them to the refrigerator for at least 3 hours (or overnight). This extended chill time prevents the cookies from spreading too much during baking and helps them maintain their shape for the peach filling.

In a small bowl, mix together the 1/2 cup granulated sugar, 2.5 tsp cinnamon, and 1/4 tsp nutmeg—set aside. In another small bowl, combine the melted and cooled butter with the light brown sugar, flour, cornstarch, 1 tsp cinnamon, and salt until crumbly. For the glaze, whisk together the sifted icing sugar, 1/2 tsp vanilla extract, a pinch of cinnamon, and milk until smooth and pourable. Set both mixtures aside until ready to use.

Preheat your oven to 350°F. Remove the cookie dough balls from the refrigerator and roll each one in the cinnamon-sugar mixture from Step 4. Place them on a lined baking sheet and bake for 11-13 minutes until the edges are set but the centers are still slightly soft. Remove from the oven and immediately press a small indent into the center of each warm cookie using your thumb or the back of a spoon. Let the cookies cool for 5 minutes on the baking sheet, then transfer to a wire rack to cool completely.

Once the cookies have cooled, spoon a generous amount of the peach filling from Step 1 into the center indent of each cookie. Top the filling with a small handful of the streusel topping from Step 4, pressing gently so it adheres. Drizzle the glaze from Step 4 over the top of each cookie. I find that warming the glaze slightly makes it drizzle more smoothly and creates a prettier presentation. Allow the glaze to set for 10-15 minutes before serving.