Preheat oven to 350°F. Line an 8x8-inch metal baking pan with parchment paper, leaving overhang on the sides.
Melt butter and chopped chocolate together in a microwave-safe bowl in 20-second intervals, stirring until smooth. Let cool slightly.
In a large bowl, whisk granulated sugar, brown sugar, eggs, egg yolk, and vanilla for 2–3 minutes until glossy and slightly thickened.
Slowly whisk in the melted chocolate mixture.
Fold in flour, cocoa powder, salt, and espresso powder until just combined. Do not overmix.
In a separate bowl, beat softened cream cheese until smooth. Add sugar, egg, and vanilla. Mix until creamy.
Spread about ¾ of the brownie batter into the prepared pan.
Spoon cheesecake filling over the brownie layer.
Dollop remaining brownie batter over the cheesecake layer.
Use a knife to gently swirl the cheesecake and brownie batter together.
Bake for 38–45 minutes, or until edges are set and the center has a slight jiggle.
Cool completely at room temperature, then refrigerate for at least 1 hour before slicing.
Slice into 16 squares using a warm, clean knife.
For the cleanest slices, chill brownies before cutting and wipe the knife between each slice. Do not overbake — brownies should come out with moist crumbs, not a clean toothpick.