In a medium bowl, whisk together the flour, cornstarch, baking soda and salt. Set aside.
In a large bowl, beat the dairy free butter, brown sugar and granulated sugar for 2 minutes until light and fluffy. Beat in the egg, egg yolk and vanilla until smooth.
Add the dry ingredients and mix on low speed just until combined. Gently fold in the white chocolate chips and most of the chopped sandwich cookies, saving about 1/2 cup for the tops.
Cover and chill the dough for 1 hour. This firms up the soft dough so the cookies bake up thick and bakery-style.
Preheat the oven to 350°F and line two baking sheets with parchment paper. Roll the dough into 2 tablespoon balls, press the reserved cookie pieces onto the tops and place them 3 inches apart.
Bake for 10-12 minutes, until the edges are lightly golden and the centers look soft. Let the cookies cool on the sheet for 10 minutes, then move them to a wire rack.