Grease a 9×13-inch baking pan with butter or nonstick spray. Line it with parchment paper, leaving an overhang on two sides to lift the brownies out later, then lightly grease the parchment. Preheat the oven to 350°F (175°C).
butter or nonstick spray, parchment paper
In a large mixing bowl, whisk together the melted, slightly cooled butter, granulated sugar, and brown sugar until thick and shiny, 1–2 minutes by hand. Add the eggs and vanilla and whisk again until the mixture is smooth and slightly lightened in color for a glossy brownie top.
1 cup unsalted butter, 1 3/4 cups granulated sugar, 1/2 cup light brown sugar, 4 large eggs, 2 teaspoons vanilla extract
In a separate bowl, whisk together the flour, cocoa powder, salt, and baking powder, breaking up any cocoa lumps. Gently fold the dry mixture into the wet ingredients with a spatula just until combined. Do not overmix.
1 cup all-purpose flour, 3/4 cup unsweetened cocoa powder, 1/2 teaspoon fine sea salt, 1/2 teaspoon baking powder
Fold in the chocolate chips or chopped chocolate. The batter will be thick and glossy. Scrape about three-quarters of the brownie batter into the prepared pan and spread into an even layer.
1/2 cup semisweet chocolate chips or chopped chocolate
In a medium bowl, beat the softened cream cheese until completely smooth and lump-free. Add the 1/4 cup granulated sugar and beat again. Mix in the egg yolk, vanilla, and 1 tablespoon of sour cream or Greek yogurt. If the mixture seems too thick, add up to 1 more tablespoon sour cream until it is creamy and spreadable.
8 oz cream cheese, 1/4 cup granulated sugar, 1 large egg yolk, 1/2 teaspoon vanilla extract, 1-2 tablespoons sour cream or plain Greek yogurt
Spoon the cream cheese mixture over the brownie layer in 8–10 large dollops. Drop the remaining brownie batter in between and around the cream cheese dollops. Using a butter knife or skewer, gently drag through the batter in long, curvy motions to create a marbled effect. Make only a few passes so the colors stay distinct.
Place the pan on the center rack and bake for 30–38 minutes. Start checking around 28 minutes. The brownies are done when the edges are set and pulling away slightly, the cream cheese swirl looks set (not jiggly in the center), and a toothpick inserted into a brownie portion comes out with a few moist crumbs but no wet batter.
Set the pan on a wire rack and let the brownies cool to room temperature, 1–2 hours. For the cleanest slices and best cheesecake texture, chill in the refrigerator for another 1–2 hours before cutting.
Use the parchment overhang to lift the slab from the pan. Transfer to a cutting board and slice into 16 large squares or up to 24 smaller pieces, wiping the knife clean between cuts for neat edges.