Preheat your oven to 350°F (175°C). Grease or line the baking dish with parchment paper for easy removal.
In a bowl, mix the crushed graham crackers, ¼ cup sugar, and melted butter until well combined. Press this mixture into the bottom of the prepared baking dish to form a crust.
Bake the crust in the preheated oven for about 10 minutes, then remove and let it cool slightly.
In a large mixing bowl, using an electric mixer, combine the softened cream cheese and ½ cup sugar until smooth and creamy.
Add the eggs one at a time, mixing well after each addition. Then add vanilla extract and sour cream, mixing until combined.
Fold in the pureed blackberries and chopped pistachios gently using a rubber spatula until just combined.
Pour the cheesecake filling over the cooled crust and spread it out evenly.
Bake in the oven for 30-35 minutes until the center is set but still slightly jiggly.
Remove from the oven and let it cool at room temperature for about 30 minutes, then refrigerate for at least 2 hours to fully set.
Before serving, top with whole blackberries and sprinkle the remaining chopped pistachios on top.