Combine blackberries, sugar, lemon juice, and lemon zest in a saucepan over medium-low heat. Cook, stirring often, until berries fall apart, 8-10 minutes. Strain mixture to remove seeds and return the sauce to the saucepan.

Mix cornstarch and water in a small bowl, then stir into the blackberry sauce. Cook until slightly thickened, about 1-2 minutes. Cool completely for 20 minutes.

Preheat the oven to 350°F (175°C). Line an 8x8-inch square pan with parchment paper.

Whisk melted butter, brown sugar, and salt together. Add egg, egg yolk, and vanilla extract, then mix in cocoa powder, flour, and baking soda to form brownie batter. Spread batter evenly in the prepared pan.

Beat cream cheese, sugar, salt, and nutmeg until smooth. Add egg white, lemon juice, and vanilla, followed by heavy cream. Pour cheesecake layer over the brownie layer and smooth evenly.

Spread dollops of cooled blackberry sauce over the cheesecake layer. Swirl the sauce and cheesecake layer with a knife for a marbled effect.

Bake for 55-60 minutes until the edges are puffed and golden. Cool to room temperature, then refrigerate for at least 4 hours or overnight.