For the Blackberry Sauce
Place blackberries and sugar in a pan and heat until the blackberries release their juices.

Combine cornstarch, juice, and water in a small bowl. Stir together until smooth.

Add slurry to pan and stir to mix completely. Bring mixture to boil for a few minutes. Allow to cool to room temperature.

For the Brownie Batter
Preheat oven to 350F. Grease an 8x8" square metal baking pan.

In a saucepan set over low heat, melt the butter, then add the sugar and stir to combine. Return the mixture to the heat briefly, just until it's hot, but not bubbling; it'll become shiny looking as you stir it. Heating this mixture a second time will dissolve more of the sugar, which will yield a shiny top crust on your brownies.

Transfer the sugar mixture to a medium-sized mixing bowl, if you've heated it in a saucepan. Stir in the cocoa, salt, baking powder, espresso powder, and vanilla.

Whisk in the eggs, stirring until smooth. Add the flour, again stirring until smooth.

For the Cheesecake
Beat the cream cheese in the bowl of a mixer until smooth. Add remaining ingredients and beat until well combined.

Add the blackberry sauce and stir until distributed.

To Assemble the Brownies
Spread about 2/3 of chocolate batter into prepared pan, and spoon cheesecake batter over. Dollop remaining brownie batter over cheesecake batter. Swirl the batters together by running a butter knife back and forth through the pan.

Bake for 40 minutes, or until center is set. Cool completely on wire rack and chill before cutting and serving.