Prepare Blackberry Puree: In a medium saucepan, combine 2 cups blackberries, 1/4 cup granulated sugar, and 1/4 cup water over medium-high heat. Cook for about 8-10 minutes, stirring occasionally, until soft and bubbly. Strain the mixture and let cool.

Brownie Base: Preheat oven to 325°F (163°C) and line an 8"x8" baking dish with parchment. Melt 12 tablespoons unsalted butter in a microwave-safe bowl. Stir in 1 1/2 cups sugar, 2 eggs, 2 teaspoons vanilla, and 1/2 teaspoon salt. Fold in 3/4 cup cocoa powder and 1/2 cup flour, then spread in the baking dish.

Cheesecake Layer: Beat together 8 ounces cream cheese, 1/4 cup Greek yogurt, 1 egg, 1/4 cup sugar, and 1/2 teaspoon salt until smooth. Pour over brownie batter.

Add Swirl: Drizzle the cooled blackberry puree over the cheesecake layer. Use a toothpick to swirl the puree without disturbing the brownie base.

Bake: Bake for approximately 1 hour. The cheesecake should be set but slightly jiggly. If the toothpick comes out too wet, bake an additional 5-10 minutes.

Cool and Serve: Let cool on a wire rack. Refrigerate for at least 2 hours, then slice into 9 squares before serving.