Step 1: Prepare the Pistachio Crust
1. Preheat Oven: Preheat your oven to 350°F (175°C). Grease a 9-inch springform pan.
2. Mix Crust Ingredients: In a medium bowl, combine the finely chopped pistachios, graham cracker crumbs, sugar, and melted butter. Mix until well combined.
3. Press into Pan: Press the mixture firmly into the bottom of the prepared springform pan to form an even layer.
4. Bake: Bake the crust in the preheated oven for 8-10 minutes, until lightly golden. Remove from the oven and let it cool.

Step 2: Make the Pistachio Cheesecake Filling
1. Beat Cream Cheese: In a large mixing bowl, beat the softened cream cheese until smooth and creamy.
2. Add Sugar and Vanilla: Gradually add the sugar and vanilla extract, beating until well combined.
3. Incorporate Eggs: Add the eggs one at a time, mixing on low speed after each addition until just combined. Be careful not to overmix.
4. Add Sour Cream and Pistachio Paste: Gently fold in the sour cream, pistachio paste, and cornstarch until the mixture is smooth and well combined.
5. Pour Filling: Pour the cheesecake filling over the cooled pistachio crust in the springform pan.

Step 3: Bake the Cheesecake
1. Bake: Bake in the preheated oven for 60-70 minutes, or until the center is set but slightly jiggly. Turn off the oven, crack the door, and let the cheesecake cool in the oven for about 1 hour to prevent cracking.
2. Cool Completely: Remove from the oven and let it cool to room temperature. Refrigerate for at least 4 hours, or overnight, before serving.

Step 4: Prepare the Raspberry Topping
1. Cook Raspberries: In a small saucepan over medium heat, combine the raspberries, sugar, lemon juice, cornstarch, and water. Cook while stirring gently until the mixture thickens and the raspberries break down (about 5-7 minutes). Remove from heat and let it cool.
2. Strain (Optional): If you prefer a smoother sauce, strain the mixture through a fine mesh sieve to remove the seeds.

Step 5: Assemble and Serve
1. Remove Cheesecake from Pan: Carefully run a knife around the edge of the springform pan to loosen the cheesecake, then remove the sides.
2. Top with Raspberry Sauce: Spoon the raspberry topping over the cheesecake, spreading it evenly.
3. Garnish: Optionally, garnish with additional fresh raspberries and chopped pistachios.
4. Slice and Serve: Cut into wedges and serve chilled.