Cook blackberry puree by combining all ingredients in a small saucepan over medium-high heat for 8-10 minutes. Use a whisk or spoon to help break down the blackberries. Once cooked, place a fine mesh strainer over a small bowl and, using a spatula, squeeze through all of the liquid and dispose of the solids. Let cool to room temperature.
Preheat oven to 325°F. Line an 8″x8″ baking dish with parchment paper and set aside. In a microwave-safe mixing bowl, melt the butter. Stir in sugar, eggs, vanilla extract, and salt. Once combined, fold in cocoa powder and flour. Pour the brownie batter into the baking dish, spreading evenly to the edges.
Combine all cheesecake ingredients in a standing mixer fitted with a whisk attachment. Cream for 2-3 minutes on medium-high speed. Pour over brownie batter, spreading evenly to the edges. Drizzle the blackberry puree on top of the cheesecake batter and swirl using a fork, knife, or toothpick. Try to swirl only into the cheesecake batter, not the brownie batter.
Bake for 1 hour, or until the cheesecake begins to turn golden and a knife inserted in the center of the brownies comes out mostly clean. Place in fridge and allow to cool for at least 2 hours but up to overnight. Cut into 9 large squares. Store brownies in the fridge in an airtight container for up to 5 days.