In a medium saucepan over medium heat, add diced peaches, granulated sugar, light brown sugar, lemon juice, cinnamon, and nutmeg. Cook for about 6 minutes, stirring occasionally until bubbly and fragrant. Mix corn starch and water in a small bowl, then add to the peach mixture. Cook for an additional minute, remove from heat and stir in vanilla extract. Cool completely in the refrigerator.

In a large mixing bowl, cream the softened butter with an electric mixer. Gradually add granulated and brown sugars, mixing until well combined. Add vanilla extract and eggs, then mix until just incorporated. Slowly add the dry ingredients, stirring until a soft dough forms. Chill the dough in the refrigerator for 30 minutes.

Scoop the chilled dough into golf ball-sized portions. Return the dough balls to the refrigerator or freezer as needed to keep firm.

Preheat oven to 350°F (175°C). Melt butter for the crumble and combine with the remaining streusel ingredients in a bowl until crumbly. Spread crumb mixture on a parchment-lined baking sheet and bake for 12-14 minutes until golden brown. Remove from the oven and let cool.

Preheat oven to 350°F (175°C). Line cookie sheets with parchment paper. Roll chilled dough balls in a mixture of granulated sugar and cinnamon before placing on prepared sheets. Bake for 11-13 minutes or until cookies are set. Use a round cutter to reshape the cookies while still warm. Let them cool on the sheets for a few minutes.

To add a glaze, whisk together glaze ingredients until smooth and adjust consistency with milk as needed.

Spoon cooled peach filling into the center of each cookie. Sprinkle with crumble topping and drizzle with glaze if desired. Serve warm and store leftovers in an airtight container in the fridge for 4-5 days.