Preheat oven to 325°F. Line an 8-inch square baking pan with aluminum foil, making sure to leave some overhang for easy removal, and set aside.

Add the graham crackers to a food processor or blender and process until you have fine crumbs. Transfer the crumbs to a large mixing bowl, add in the granulated sugar and melted butter, and mix until all of the crumbs are moistened. Scoop into the prepared baking pan and firmly press down into one even layer.

Bake at 325°F for 10 minutes. Remove from the oven and transfer to a wire rack to cool.

Add the blackberries to a food processor or blender and process until smooth. Strain the blackberry puree through a fine mesh strainer into a small bowl and discard any large pieces. Add the granulated sugar to the blackberry mixture and mix until well combined.

In the bowl of a stand mixer fitted with the paddle attachment, or in a large mixing bowl using a hand-held mixer, beat the cream cheese until smooth. Next, add in the sour cream, granulated sugar, and vanilla extract and continue mixing until well combined and smooth. Add in the eggs one at a time and mix on low until just combined.

Pour the cheesecake filling over the prepared crust and spread it out into one even layer. Spoon the blackberry mixture in different spots on top of the cheesecake filling and gently swirl with a knife.

Bake at 325°F for about 30-35 minutes or until the cheesecake has set up.

Remove from the oven to a wire rack to cool completely. Once cooled, cover tightly and transfer to the refrigerator for 3-4 hours, or overnight, or until fully chilled. Slice into bars and enjoy.