1. Prepare the Brownie Batter: Gently melt the butter and chopped semi-sweet chocolate in a medium saucepan over medium heat, stirring until smooth (about 5 minutes). Alternatively, use a microwave-safe bowl, heating in 20-second bursts and stirring after each until melted. Transfer to a large mixing bowl and let it cool slightly for about 10 minutes.
2. Add Wet Ingredients to Brownies: Whisk the granulated sugar into the cooled chocolate and butter mixture. Incorporate the eggs one at a time, whisking well after each addition until the batter is smooth. Stir in the pure vanilla extract.
3. Combine Dry Brownie Ingredients: Carefully fold in the flour, cocoa powder, espresso powder, and salt until just combined. Don't overmix. Set this rich brownie batter aside.
4. Preheat Oven & Prep Pan: Adjust your oven rack to the lower-third position and preheat to 350°F (177°C). Line a 9-inch square baking pan with parchment paper, leaving some overhang on the sides for easy lifting later.
5. Make the Cheesecake Filling: In a separate large bowl, using an electric mixer, beat the softened cream cheese on high speed for about 1 minute until it's super smooth and creamy. Reduce to medium-high speed, add the granulated sugar, and beat until just combined. Mix in the egg and vanilla extract until everything is fully incorporated. Finally, gently fold in the white chocolate chips with a spatula.
6. Assemble the Layers: Spread approximately half of your thick brownie batter evenly into the prepared baking pan. Next, carefully spread all of the creamy cheesecake batter over the brownie layer in an even sheet. Then, dollop and gently spread the remaining brownie batter on top of the cheesecake. Don't worry if it's not perfectly covered or if some cheesecake peeks through – it will all bake beautifully! If desired, sprinkle the optional chocolate chips on top.
7. Bake to Perfection: Bake for 35-38 minutes. You'll know they're ready when the edges just begin to pull away from the pan and a toothpick inserted into the center comes out with only a few moist crumbs. Keep an eye on them, as oven times can vary.
8. Cool and Chill Thoroughly: Let the brownies cool completely in the pan on a wire rack for about 45 minutes. For that perfect cheesecake texture and clean cuts, cover the pan and refrigerate for at least 2 hours, or even overnight. The cheesecake layer needs to be firm and cold before slicing.
9. Slice and Serve: Once chilled, use the parchment paper overhang to lift the entire batch of brownies out of the pan. Cut into neat squares using a sharp knife, wiping the knife clean after each cut for the best results. Enjoy your delicious homemade treat!