In a large bowl, beat the butter, 1/2 cup cookie butter, brown sugar and granulated sugar with a mixer on medium speed for 2 minutes, until light and fluffy. Beat in the milk and vanilla until smooth.
In a separate bowl, whisk together the flour, cinnamon, baking soda and salt. Add to the butter mixture and mix on low just until combined, then fold in the crushed cookies.
Cover and chill the dough for 45 minutes. Cookie butter softens quickly in the oven, so this step keeps the cookies thick and chewy.
Preheat the oven to 350°F and line two baking sheets with parchment paper. Scoop the dough into 2 tbsp balls and place them 2 inches apart.
Bake for 10-12 minutes, until the edges are golden and the centers are still soft. Right out of the oven, gently press a cookie half into the top of each one.
Cool on the baking sheet for 10 minutes, then transfer to a wire rack to cool completely.
Microwave the 1/4 cup cookie butter for 15-20 seconds, until pourable, and drizzle it over the cooled cookies.Let the drizzle set for 15 minutes before serving. Makes about 18 cookies.