Preheat the oven to 325°F (160°C). Grease the 9-inch springform pan and line the bottom with parchment paper.
In a mixing bowl, combine graham cracker crumbs, ⅓ cup sugar, and melted butter. Mix until well combined.
Press the crumb mixture evenly into the bottom of the prepared springform pan. Bake in the preheated oven for 10 minutes. Remove and let cool.
In another bowl, beat the softened cream cheese until creamy. Gradually add 1 cup sugar, beating until smooth.
Add eggs one at a time, mixing on low speed after each addition. Stir in vanilla extract and sour cream until fully combined.
Pour the cheesecake filling over the cooled crust. Tap the pan on the counter to remove air bubbles.
Bake for 50-60 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 slowly.
Remove the cheesecake from the oven and let it cool to room temperature before refrigerating for at least 4 hours or overnight.
To prepare the berry topping, in a saucepan, combine mixed berries, ¼ cup sugar, and lemon juice. Cook over medium heat until the berries start to break down (about 5 minutes).
Stir in the cornstarch-water mixture and cook until the sauce thickens, about 2 minutes. Remove from heat and let cool.
Once the cheesecake is chilled, top it with the berry mixture before serving.