Make the Cookie Dough Batter: In a saucepan, brown 10 tbsp of butter. Turn heat to low, whisk in sugars for 1-2 minutes. Remove from heat, cool for 5 minutes. Whisk in egg, egg yolk, vanilla, and dissolved coffee until creamy. In a separate bowl, mix flour, salt, baking powder, baking soda, and chopped chocolate. Fold dry ingredients into wet ingredients until just combined. Do not overmix. Set aside.
Make the Brownie Batter: Preheat oven to 350°F (175°C) and line a 9x9 pan with parchment paper. Brown the other 10 tbsp of butter. Pour the hot butter over the 4 oz of chopped chocolate in a heatproof bowl. Let sit for 2-3 minutes, then stir until smooth. Whisk in the dissolved coffee and sifted cocoa powder.
Combine Brownie Batter: In a large bowl, whisk eggs and sugar for 2-3 minutes until pale and fluffy. Pour the warm chocolate mixture into the eggs and whisk to combine. Sift in the flour and salt, folding gently. Finally, fold in the remaining 3 oz of chopped chocolate.
Assemble the Brookies: Spread about ⅓ of the brownie batter on the bottom of the prepared pan. Dollop chunks of the cookie dough on top, leaving gaps. Spoon the remaining brownie batter into the gaps. Place any remaining cookie dough on top for a patchwork effect. Gently flatten with a spatula.
Bake: Bake for 32-35 minutes. The edges should be set, but a toothpick in the center should come out with moist, sticky crumbs (not wet batter). It's better to underbake than overbake.
Cool and Serve: Let the brookies cool completely in the pan for at least 30 minutes, or up to 2 hours for the best texture. Use the parchment handles to lift them out, then cut with a sharp knife. Enjoy!