With an electric mixer, cream butter and sugars in a small mixing bowl. Add vanilla and egg; cream again until smooth. Add the flour, baking soda, and salt. Stir gently until incorporated. Stir in chocolate chips. Store in refrigerator until ready to use.
Preheat oven to 350 degrees. Prepare brownie batter as directed. Pour batter into a glass 8×8 or 9×9 baking dish. Scatter pieces of cookie dough over the top of the brownie batter. In my 8×8 square pan, I was able to get about 2/3 of the cookie dough to fit over the brownies and I just saved the rest for regular cookies. Press gently on the dough just until it starts to sink down into the batter a little bit.
Bake for 35-40 minutes (35 for gooey, 40 or more for fully baked), covering with foil if you need to keep the cookies from getting too brown on top. Let stand for 15-20 minutes and cut into pieces for serving.