Preheat your oven to 350°F (175°C) and line your 8×8 inch baking pan with parchment paper, leaving an overhang on the sides for easy removal.
In a medium bowl, whisk together the melted butter and 1 cup of sugar until smooth. Add the 2 eggs and vanilla extract, beating until well combined.
Sift the flour, cocoa powder, and salt into the wet mixture. Fold gently with a spatula until just combined, being careful not to overmix. Spread this brownie batter into the prepared pan.
In a separate clean bowl, beat the softened cream cheese with the 1/4 cup of sugar and the egg yolk until the mixture is light and airy.
Drop spoonfuls of the cheesecake mixture over the top of the chocolate batter. Use a knife or a skewer to gently swirl the two mixtures together, creating a marbled effect.
Scatter the fresh blackberries evenly over the top, pressing them slightly into the batter.
Bake for 30 to 35 minutes. The edges should be set, and a toothpick inserted into the brownie portion (avoiding the berries) should come out with just a few moist crumbs attached.
Allow the pan to cool completely on a wire rack before lifting the brownies out by the parchment paper and slicing into squares.