Preheat your oven to 325°F (160°C).
In a medium mixing bowl, combine the graham cracker crumbs, melted butter, and ¼ cup sugar to form the crust mixture. Mix until evenly combined.
Press the crumb mixture firmly into the bottom of the springform pan to create an even layer. Bake in the preheated oven for 10 minutes, then remove and allow to cool.
In a large bowl, beat the softened cream cheese and 1 cup of sugar with an electric mixer until smooth and creamy.
Add the eggs, one at a time, mixing on low speed after each addition until just combined. Stir in the vanilla extract.
Gently fold in the 2 cups of blackberries into the cream cheese mixture.
Pour the filling over the cooled crust in the springform pan. Bake for 40 minutes or until the center is set but still slightly jiggly.
Turn off the oven and leave the cheesecake inside for an additional hour to cool gradually.
While the cheesecake cools, prepare the topping. In a small saucepan, combine the 1 cup of blackberries, ¼ cup sugar, cornstarch, and water. Cook over medium heat, stirring gently until the mixture has thickened and the blackberries have softened (about 5 minutes). Let cool.
Once the cheesecake is completely cooled, remove it from the springform pan and drizzle the blackberry topping over the top. Chill in the refrigerator for at least 4 hours before serving.