Step 1: Preheat your oven to 350°F (175°C). Line an 8x8 inch baking pan with parchment paper, leaving an overhang on two sides for easy removal.
Step 2: Melt the butter and chopped dark chocolate together in a heatproof bowl set over a saucepan of simmering water (double boiler method), or in a microwave-safe bowl in 30-second intervals, stirring until smooth. Let cool slightly.
Step 3: In a large bowl, whisk together the cooled chocolate mixture and 1 cup of granulated sugar until well combined. Beat in the 2 large eggs one at a time, then stir in 1 teaspoon of vanilla extract until smooth.
Step 4: In a separate small bowl, whisk together the flour, cocoa powder, and salt. Gradually add the dry ingredients to the wet ingredients, mixing just until no streaks of flour remain. Do not overmix. Pour about two-thirds of the brownie batter into the prepared pan and spread evenly. Reserve the remaining brownie batter.
Step 5: In a medium bowl, beat the softened cream cheese with 1/4 cup of granulated sugar until smooth and creamy. Beat in the egg yolk and 1/2 teaspoon of vanilla extract until fully combined.
Step 6: Carefully dollop the cream cheese mixture over the brownie batter in the pan. Then, drop spoonfuls of the remaining reserved brownie batter over the cheesecake layer.
Step 7: Using a knife or skewer, gently swirl the brownie and cheesecake layers together to create a marbled effect.
Step 8: Bake for 30-35 minutes, or until the edges are set and a toothpick inserted into the brownie portion comes out with moist crumbs (not wet batter). The cheesecake layer might still have a slight jiggle.
Step 9: Remove from the oven and let cool completely on a wire rack. For cleaner slices, chill in the refrigerator for at least 2 hours before cutting and serving.