Preheat oven to 350 degrees F. Spray a 9x9-inch baking pan with nonstick spray and line with parchment paper, then spray paper. Prepare brownie batter according to package directions and stir in mini chocolate chips. Bake and let cool completely.
In a medium bowl, stir to dissolve espresso powder in boiling water. Set aside to cool.
In the bowl of stand mixer with the whisk attachment or with a hand held mixer with whisk attachments, whip vanilla pudding mix with 2 cups cold milk and mascarpone on medium speed until smooth and thickened, about 2 minutes. Use a rubber spatula to fold in 1 cup Cool Whip.
Spoon half of the mascarpone pudding onto the cooled brownies. Working one at a time, dip the ladyfinger cookies in the espresso, turning to coat, and place in a single layer on top of the pudding. Top cookies with remaining mascarpone pudding and smooth to cover. Top Tiramisu brownies with remaining Cool Whip and refrigerate until set, at least 30 minutes and up to 4 hours.
To serve, use the parchment paper overhang to lift the cake from the pan, dust with cocoa powder and cut into squares.