Make The Blackberry Swirl: In a small saucepan over medium heat, combine the blackberries, sugar, and lemon juice. Cook for about 5 minutes until the berries break down and thicken into a sauce-like consistency.
Prepare The Brownie Batter: Preheat your oven to 325°F (163°C). Grease an 8x8-inch baking pan. Melt butter and semi-sweet chocolate together until smooth; allow to cool slightly. Whisk in granulated sugar, eggs, and vanilla extract.
Combine Dry Ingredients For The Brownie Layer: Sift together flour, cocoa powder, and salt. Fold these dry ingredients into your wet mixture until just combined.
Make The Cheesecake Swirl: Beat softened cream cheese with granulated sugar, one egg, and vanilla extract until smooth. Optionally, divide this mixture and incorporate melted dark chocolate into one part.
Assemble And Bake Your Brownies: Spread brownie batter into prepared pan followed by dollops of cheesecake mix & blackberry sauce. Bake for about 35-40 minutes.