In the bowl of a stand mixer, combine the butter, sugars and extracts. Beat until light and fluffy, about 2 minutes. With mixing speed on low, add egg and mix until combined.
In a medium size mixing bowl, whisk together the flour, cocoa powder, baking soda, and salt. Alternate adding the dry ingredients and the milk, mixing just until combined.
Place dough in the refrigerator and chill for 1 hour.*
Preheat oven to 350 degrees. Line baking sheets with parchment paper or baking mats, set aside.
Using a standard size cookie scoop (1.5 tbsp.), create balls of dough. Divide the balls into half and sandwich the Joe-Joe's in between so that half of the dough is on top and half is on the bottom. Pat to seal the cookie inside. Place on prepared cookie sheets and sprinkle with peppermint chips.
Place in the oven to bake for 10-12 minutes. The cookies should be soft but the edges should be set. Remove from the oven and allow to cool for 3-4 minutes before transferring to a wire rack to cool completely.
Notes
*Dough must be chilled in order to handle it. If not, it will be too sticky and will be all over your hands!