For blackberry sauce:

Combine frozen blackberries, sugar, lemon juice, and lemon zest in a small saucepan set over medium-low heat. Cook, stirring often, until berries fall apart, 8 to 10 minutes. Remove mixture from heat, and pour into a fine mesh sieve set over a bowl. Use the back of a wooden spoon or spatula to press berry mixture through the sieve, extracting as much juice as possible. Discard seeds. Pour sauce back into saucepan and bring to a simmer over low heat.

Stir cornstarch and water together in a small bowl until thoroughly combined; stir cornstarch slurry into blackberry sauce and continue to cook, stirring constantly, until slightly thickened, 1 to 2 minutes. Remove from heat, and allow blackberry sauce to cool completely, about 20 minutes. The sauce can be refrigerated to speed up cooling.

For brownie:

Preheat oven to 350 degrees F (175 degrees C). Line an 8×8-inch square pan with enough parchment paper to have overhang on all sides.

Whisk 1/2 cup melted butter, 1 cup brown sugar, and 1/2 teaspoon salt together in a large bowl until thoroughly combined. Whisk in 1 large egg, 1 large egg yolk, and 2 teaspoons of vanilla extract until thoroughly incorporated. Add in 1/2 cup cocoa powder, 1/2 cup all-purpose flour, and 1/4 teaspoon of baking soda and stir until just combined. Batter will be somewhat thick. Spread brownie batter into the prepared pan in a smooth, even layer.

For cheesecake:

Beat 8 ounces of cream cheese, 1/3 cup white sugar, 1 pinch salt, and 1 pinch nutmeg together in a bowl with an electric mixer until smooth and creamy. Add in 1 large egg white, 2 teaspoons lemon juice, and 1 teaspoon of vanilla extract and mix until thoroughly incorporated. Pour in 3/4 cup heavy cream and mix until combined. Pour cheesecake layer over brownie layer and smooth into an even layer.

Place dollops of the cooled blackberry sauce over the cheesecake layer. Use a knife to swirl only the cheesecake layer and blackberry sauce together; taking care not to go into the brownie layer. Swirl as many or as few times as needed to get the desired marbled effect.

Bake in the preheated oven until the center of the bars are puffed and mostly set, 55 to 60 minutes. The edges will bake up taller than the center, and will begin to turn golden brown when they are done. Allow bars to cool to room temperature in the pan, then refrigerate for at least 4 hours to overnight. Cut into 16 bars before serving.