In a medium bowl, beat cream cheese, granulated sugar, and vanilla extract on medium speed until smooth, about 1-2 minutes.
Scoop 20 dollops (~2 to 2 1/2 teaspoons each) of cream cheese mixture onto a parchment-lined baking sheet and freeze uncovered for 45 to 60 minutes until solid.
Sift together flour, cornstarch, baking powder, baking soda, and salt in a medium bowl and set aside.
In a stand mixer bowl, cream butter, dark brown sugar, and granulated sugar on medium speed until fluffy, about 1-2 minutes.
Add eggs one at a time, mixing well after each addition, then blend in vanilla extract.
Reduce mixer speed to low and gradually add the flour mixture, mixing just until combined.
Gently fold in finely chopped Oreo cookies evenly throughout the dough.
Cover dough with plastic wrap and refrigerate for 25-30 minutes to firm.
Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
Remove frozen cream cheese dollops from freezer. Using a 3-tablespoon scoop, form cookie dough balls and press a small indent in the center.
Place a frozen cream cheese dollop in the indent and carefully wrap dough around it to seal, leaving a small visible gap.
Place each cookie on baking sheets about 2-3 inches apart.
Bake one sheet at a time for 13-15 minutes, until edges are lightly golden and centers are set. Do not overbake.
Immediately press the roughly chopped Oreo pieces onto the top of warm cookies.
Cool cookies on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
Repeat baking and decorating until all cookies are baked.