Prepare Blackberry Puree:
In a saucepan, combine the blackberries, sugar, and water. Cook over medium-high heat for about 8-10 minutes, stirring occasionally to break down the blackberries.
Pour the mixture through a fine mesh strainer to remove the solids, leaving only the smooth blackberry liquid.
Let the puree cool to room temperature.
Make the Brownie Layer:
Preheat the oven to 325°F (163°C). Line an 8×8-inch baking dish with parchment paper or grease it lightly.
Melt the butter in a microwave-safe bowl in 30-second intervals.
Stir in sugar, eggs, vanilla, and salt until well combined.
Sift together the cocoa powder and flour, then fold them into the wet ingredients until incorporated.
Pour the brownie batter into the prepared baking dish, spreading evenly.
Prepare the Cheesecake Layer:
In a stand mixer or using a hand mixer, beat together the cream cheese, Greek yogurt, egg, sugar, and salt until smooth and creamy.
Pour the cheesecake mixture over the brownie batter, spreading it evenly across the top.
Swirl the Blackberry Puree:
Drop spoonfuls of the blackberry puree over the cheesecake layer.
Use a toothpick or knife to swirl the puree into the cheesecake mixture, making sure not to disturb the brownie base.
Bake:
Bake for 1 hour, or until the cheesecake layer is slightly golden, and a knife inserted into the center comes out clean with just a little cheesecake residue.
Allow the brownies to cool to room temperature before refrigerating for at least 2 hours (or overnight) to set.
Serve:
Once chilled, cut into squares and serve.