Melt the Chocolate Mixture: Melt the butter and coarsely chopped semi-sweet chocolate together in a medium saucepan over medium heat, stirring constantly for about 5 minutes. Alternatively, place them in a microwave-safe bowl and microwave in 20-second intervals, stirring after each interval. Remove from the heat and transfer to a large mixing bowl. Let the mixture cool slightly for 10 minutes. Prepare the Baking Pan: Position the oven rack in the lower third of the oven and preheat the oven to 350°F (177°C). Line the bottom and sides of a 9-inch square baking pan with parchment paper, leaving enough overhang on all sides to lift out the chilled brownies later.
Finish the Brownie Batter: Whisk the granulated sugar into the slightly cooled chocolate mixture. Add the eggs one at a time, whisking until smooth after each addition. Whisk in the vanilla extract. Gently fold in the flour, cocoa powder, espresso powder, and salt until combined. Set the brownie batter aside. Smooth the Cream Cheese: Place the softened cream cheese in a large bowl. Using a handheld mixer or stand mixer fitted with a paddle or whisk attachment, beat it on high speed for 1 minute, until completely smooth and creamy.
Make the Cheesecake Batter: Reduce the mixer to medium-high speed and beat the sugar into the cream cheese until combined. Add the egg and vanilla extract, then beat until incorporated. Fold in the white chocolate chips using a silicone spatula or wooden spoon.
Build the First Brownie Layer: Spread about half of the brownie batter across the prepared pan. Because the batter is thick, carefully spread it toward all four corners to create an even layer.
Add the Cheesecake Layer: Spoon all of the cheesecake batter over the brownie layer and spread it into a thick, even layer.
Complete the Layers: Spread the remaining brownie batter over the cheesecake layer as evenly as possible. It is fine if some cheesecake mixture shows through or the brownie batter does not completely reach every corner. Sprinkle the optional semi-sweet chocolate chips over the top.
Bake the Brownies: Bake at 350°F (177°C) for 35–38 minutes, or until the brownies begin pulling away from the edges of the pan.A toothpick inserted into the center should come out with only a few moist crumbs. Begin checking after 30 minutes because oven performance can vary. Cool the Brownies: Place the baking pan on a wire rack and allow the brownies to cool in the pan for 45 minutes.
Chill the Cheesecake Layer: Cover the pan and refrigerate the brownies for at least 2 hours and up to 1 day. The cheesecake layer needs to be thoroughly cold before cutting.
Slice the Brownies: Once chilled, use the parchment-paper overhang to lift the brownies from the pan. Cut into squares using a very sharp knife, wiping the blade clean with a paper towel after each cut for neater slices.
