Make the Blackberry Puree: Start by combining the blackberries, 1/4 cup of sugar, and 1/4 cup of water in a saucepan. Cook over medium-high heat for about 8 to 10 minutes. Stir gently and break down the blackberries as they cook.

Strain the Puree: After cooking, strain the mixture through a fine mesh strainer to remove the seeds. Let the puree cool while you prepare the brownies.

Preheat the Oven: Preheat your oven to 325°F (163°C). Line an 8×8 inch baking dish with parchment paper for easy removal.

In a mixing bowl, melt the butter. Once melted, stir in the 1 1/2 cups of sugar, 2 eggs, vanilla extract, and 1/2 teaspoon of salt. Mix well until combined.

Fold in the cocoa powder and flour gently. Mix until just combined. Do not overmix to keep the brownies fudgy.

Pour the brownie batter into the prepared baking dish. Use a spatula to spread it evenly across the bottom.

In a separate mixing bowl, beat the softened cream cheese until smooth. Add the Greek yogurt, 1 large egg, 1/4 cup sugar, and 1/2 teaspoon salt. Beat for 2 to 3 minutes until the mixture is creamy and smooth.

Spread the cheesecake mixture evenly over the brownie batter.

Take the cooled blackberry puree and dollop it on top of the cheesecake layer. Use a knife or toothpick to swirl it gently into the cheesecake.

Place the baking dish in the preheated oven. Bake for about 1 hour, or until the cheesecake is golden and a knife inserted in the center comes out mostly clean.

Allow the brownies to cool at room temperature. Then refrigerate for at least 2 hours before serving to help the flavors meld and make slices easier to cut.