Preheat oven to 350° F. Butter an 8-inch square pan. Add a layer of parchment paper to the pan, leaving the edges long enough to lift the brownies out of the pan once cool or easier slicing. Butter the parchment paper, pressing it to adhere smoothly to the pan.
For the cookie dough, melt butter in a medium saucepan over moderate heat. Add sugars to a medium bowl and stir to combine. When the butter has just finished melting, pour it over the sugars, setting the pan aside without washing to use for the brownies. Mix butter into the sugars with a wooden spoon until well combined. Add egg yolk and vanilla extract and mix well. Stir in salt and baking soda. Add ¼ cup of the flour and stir until just combined. Add remaining flour 1 tablespoon at a time, stirring until just combined. Mix in chocolate chips, cover, and refrigerate while you prepare the brownie batter.
For the brownies, melt butter in saucepan over moderate heat. Add sugar and stir to combine. Continue heating the mixture until it is hot but not bubbling; it will become shiny as you stir it. Stir in cocoa powder and vanilla extract. Add eggs, one at a time, beating until smooth after each addition. Add salt and baking powder, stirring until combined. Add flour, ¼ cup at a time, stirring until the flour is incorporated.
Scrape the batter into prepared pan, spreading it as evenly as you can; it will be thick and glossy. Divide prepared cookie dough into 16 even pieces. Roll dough into balls and gently press them into the brownie batter. If desired, press chocolate chips into the tops of the cookie dough balls so they will show after baking.
Bake until a toothpick inserted in the center comes out with just a few clinging crumbs, about 35 minutes. Place pan on rack to cool completely.
Gently lift the cooled brownies from the pan and transfer them to a cutting board. Slice into 2-inch squares and serve.