Preheat oven to 350℉. Line a 9x9 inch baking pan with parchment paper.
Brownie layer: Combine melted butter, sugar, eggs, and vanilla in a stand mixer. In a medium bowl, combine cocoa powder, flour, espresso, and salt. Add the cocoa mixture to the butter mixture. Mix until just incorporated. Fold in the chocolate chips. Pour the brownie batter into the bottom of the prepared pan and smooth with a spatula.
Cheesecake layer: Using a stand mixer with a paddle attachment, beat the cream cheese, butter, and sugar on medium-high speed until smooth and creamy, about 2 minutes. Add the flour, egg, and vanilla and beat until smooth. Fold in the pieces of 8 cookies. Pour the cheesecake layer on top of the brownie layer and smooth out with a spatula. Top the cheesecake layer with the pieces of the remaining 4 cookies and 2 tablespoons each of dark chocolate chips and white chocolate chips.
Bake cheesecake brownies for 38 to 45 minutes, or until the edges are set and lightly browned and the center is slightly wobbly. A toothpick inserted into the center should come out mostly clean with a few crumbs. If the edges start browning early, loosely tent with foil (I tend to do this about 30 minutes into baking).
Cool the brownies in the pan on a wire rack for 1 hour before transferring to the refrigerator to finish cooling. Allow the brownies time to cool completely before cutting.