Place the sandwich cookies in a zip-top bag and crush them with a rolling pin into chunky pieces, leaving some larger bits for texture. Set aside 1/4 cup of the larger pieces for topping.
In a large bowl, beat the 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 combined.
In a separate bowl, whisk together the flour, cornstarch, baking soda and salt. Add to the butter mixture and mix on low just until combined.Fold in the crushed cookies and white chocolate chips with a spatula.
Cover and chill the dough for 30 minutes. Near the end, preheat the oven to 350°F and line two baking sheets with parchment paper.
Scoop the dough into 2 tbsp balls, place them 2 inches apart and press the reserved cookie pieces onto the tops.
Bake for 10-11 minutes, until the edges are lightly golden and the centers still look soft.
Cool on the baking sheet for 10 minutes, then transfer to a wire rack. Makes about 18 cookies.