In a medium bowl, whisk the flour, cocoa powder, baking soda and salt together. Set aside.

In a large bowl, beat the butter with a mixer on high speed until creamy. Add the brown sugar, granulated sugar, and peanut butter until combined. Beat in the egg and vanilla extract (if using), scraping down the sides and bottom of the bowl as needed.

Pour the dry ingredients into the wet ingredients. Mix until a soft dough is formed. Beat in the milk. Once combined, fold in the broken KitKat Chunky. The dough will be very thick and sticky. Cover the dough tightly with aluminum foil or plastic wrap and chill for at least 1 hour.

Preheat oven to 350°F (177°C). Line a baking sheet with parchment paper or silicone baking mat. Set aside.

Form 8 balls of dough onto prepared baking sheet. Bake for 15-17 minutes. The cookies will not have spread much, so gently press down on them with the back of a fork or spoon. Allow to cool on the cookie sheet for 5 minutes before transferring to a wire rack to cool completely.

As the cookies are cooling, melt the peanut butter. Drizzle over warm or cool cookies. Allow peanut butter to set for about 10 minutes. Store cookies in an airtight container at room temperature for up to 1 week.