Preheat the oven to 325°F (163°C).
In a food processor, pulse the shelled unsalted pistachios until finely chopped but not powdered.
In a medium bowl, combine the chopped pistachios, graham cracker crumbs, 1/4 cup granulated sugar, and melted unsalted butter. Mix until the mixture is evenly moistened.
Press the pistachio mixture firmly into the bottom of a 9-inch springform pan to form an even crust layer.
Bake the crust in the preheated oven for 10 minutes. Remove from oven and set aside to cool slightly.
In a large mixing bowl, beat the softened cream cheese with 1 cup granulated sugar until smooth and creamy, about 3 minutes.
Add the eggs one at a time, mixing well after each addition.
Mix in the sour cream and vanilla extract until fully incorporated and smooth.
Pour the cheesecake filling over the cooled pistachio crust, smoothing the top with a spatula.
Bake the cheesecake for 55 to 60 minutes, or until the edges are set and the center is slightly jiggly.
Turn off the oven and leave the cheesecake inside with the door slightly open for 1 hour to cool gradually.
Remove the cheesecake from the oven and refrigerate it for at least 4 hours or overnight to fully set.
While the cheesecake chills, prepare the raspberry topping: In a small saucepan, combine fresh raspberries, 1/4 cup granulated sugar, cornstarch, and lemon juice.
Cook over medium heat, stirring frequently, until the mixture thickens and becomes glossy, about 5 minutes. Remove from heat and let cool completely.
Once the cheesecake is fully chilled, spread the cooled raspberry topping evenly over the top.
Carefully remove the sides of the springform pan before serving.
Slice and serve chilled.