Position a rack in the center of the oven and preheat to 175°C or 350°F. Lightly grease a 9 x 13-inch light-colored metal pan, then line the bottom and long sides with parchment, leaving enough overhang to lift the slab later. Grease the exposed short sides. A metal pan conducts heat more predictably than glass, which helps the central brownie set before the perimeter becomes hard.
In a large bowl, beat the softened cookie-layer butter, brown sugar, and granulated sugar with a hand mixer on medium speed for 2 minutes, until creamy and slightly aerated but not fluffy. Beat in the egg, yolk, and vanilla until smooth. Whisk the flour, cornstarch, baking soda, and salt separately, then mix them into the butter mixture on low just until no dry flour remains. Fold in the chocolate chips and refrigerate the dough for 15 minutes.
Place the brownie-layer butter and chopped chocolate in a medium heatproof bowl. Set it over a saucepan containing about 2.5 cm or 1 inch of barely simmering water, keeping the bowl above the water. Stir until approximately three-quarters melted, then remove from the heat and continue stirring until completely smooth. Let the mixture stand for 5 minutes. Cooling it briefly prevents the eggs from heating too quickly and helps preserve a stable, glossy batter.
Whisk the granulated sugar into the warm chocolate mixture. Add the eggs one at a time, whisking vigorously for about 30 seconds after each, then whisk in the vanilla until the batter looks smooth and slightly thickened. Sift the cocoa powder, flour, and salt directly over the bowl. Fold with a flexible spatula, scraping the bottom and sides, just until the final flour streak disappears. The batter should fall from the spatula in thick, glossy ribbons.
Scrape the brownie batter into the prepared pan and spread it into an even layer, taking care to reach the corners. Remove the cookie dough from the refrigerator and flatten portions about the size of a heaped tablespoon into rough 1 cm-thick pieces. Arrange them closely over the batter, leaving narrow channels of brownie visible between the pieces. Do not press the dough into the batter. The exposed spaces allow heat and steam to move through the thicker top layer.
Bake for 32 to 36 minutes, rotating the pan from front to back after about 22 minutes if your oven browns unevenly. The cookie layer should have pale golden centers, deeper golden edges, and a dry, set surface. Insert a toothpick into an exposed brownie area near the center, avoiding chocolate chips. It should emerge with several moist crumbs but no shiny liquid batter. If the cookie top darkens early, loosely cover the pan with foil for the remaining time.
Set the pan on a wire rack and cool the brookies completely in the pan for at least 2 hours. The center continues setting as the melted fats cool, so cutting sooner can produce compressed, messy layers even when fully baked. Run a thin knife along the unlined edges, lift the slab out using the parchment, and place it on a cutting board. Use a long sharp knife to cut 24 bars, wiping the blade clean between cuts for distinct layers.