Preheat oven to 350°F (175°C). Line a 9 x 13 x 2-inch baking pan with aluminum foil or parchment paper, leaving an overhang for easy removal, and spray the lining with pan spray; set the pan aside.
In a microwave-safe bowl or over a double boiler, melt ¾ cup butter and 6 ounces chopped dark chocolate together, stirring every 30 seconds until smooth and fully melted; take care not to burn the chocolate.
Whisk 1 ½ cups granulated sugar and ½ cup packed brown sugar into the melted chocolate mixture until combined. Stir in 1 ½ teaspoon vanilla extract. Add 3 large eggs, one at a time, mixing after each addition until fully incorporated. Fold in 1 cup all-purpose flour just until combined.
Pour the brownie batter into the prepared pan and spread it into an even layer. Sprinkle ¾ cup semi-sweet chocolate chips evenly over the top of the brownie layer.
In a stand mixer fitted with the paddle attachment or using an electric hand mixer, beat 8 ounces cream cheese and ¼ cup unsalted butter on medium-high speed until fluffy (about 2 minutes). Add 1 large egg, ½ cup granulated sugar, ½ tsp salt, and ½ tsp vanilla extract; mix on medium speed until smooth. Add 1 cup creamy peanut butter and mix until fully combined and smooth.
Pour the cheesecake–peanut butter mixture over the brownie layer and smooth it into an even layer so it completely covers the brownie layer.
Bake in the preheated oven for 35–40 minutes, until the edges are set and the top is slightly golden. The center should be mostly set (a slight jiggle is okay) and not liquid.
Remove from the oven and let the bars cool completely in the pan on a wire rack. Once cooled, refrigerate the bars for at least 1 hour (or overnight) before using the foil/parchment overhang to lift them from the pan and cut into bars.