Step 1: Preheat your oven to 350°F (175°C). Lightly grease an 8×8-inch baking pan, or line it with parchment paper, leaving an overhang on two sides.
Step 2: In a medium saucepan, melt the butter over low heat. Remove from heat and stir in 1 cup granulated sugar until well combined. Beat in 2 large eggs one at a time, then stir in 1 teaspoon vanilla extract. In a separate bowl, whisk together the cocoa powder, all-purpose flour, salt, and baking powder. Gradually add the dry ingredients to the wet mixture, stirring until just combined.
Step 3: In a mixing bowl, beat the softened cream cheese with 1/3 cup granulated sugar using an electric mixer until smooth. Add the remaining 1 large egg and 1 teaspoon vanilla extract, mixing until just combined and smooth.
Step 4: Pour about half of the brownie batter into the prepared baking pan and spread it evenly. Carefully spoon the cheesecake mixture over the brownie layer, spreading it gently. Drop spoonfuls of the remaining brownie batter over the cheesecake layer. Take a knife or toothpick and gently swirl the two layers together a few times.
Step 5: Place the pan in your preheated oven and bake for 30-35 minutes, or until a toothpick inserted into the center of the brownie portion comes out with a few moist crumbs.
Step 6: Allow the bars to cool completely in the pan on a wire rack for at least 15 minutes. Transfer the cooled pan to the refrigerator and chill for at least 2 hours, or preferably overnight, before cutting into squares and serving.