Preheat your oven to 325°F (160°C).
In a mixing bowl, combine the graham cracker crumbs, melted butter, and ⅓ cup sugar. Mix until well combined, resembling wet sand.
Press the mixture firmly into the bottom of a 9-inch springform pan to form the crust. Bake for 10 minutes, then remove from the oven and let cool.
In a separate large bowl, beat the softened cream cheese with an electric mixer until smooth and creamy. Gradually add 1 cup sugar and continue mixing until well incorporated.
Add the eggs, one at a time, mixing well after each addition. Mix in the vanilla extract and sour cream until smooth.
Gently fold in the 1 cup of mixed berries into the cheesecake filling using a rubber spatula.
Pour the cheesecake filling over the cooled crust in the springform pan. Level the top with the spatula.
Bake in the preheated oven for about 60 minutes, or until the edges are set and the center is slightly jiggly. Turn off the oven and let the cheesecake cool in the oven for 1 hour to prevent cracks.
While the cheesecake is cooling, prepare the berry topping. In a small bowl, combine 1 cup mixed berries, 2 tbsp sugar, and 1 tbsp lemon juice. Gently toss to coat and let sit for 15 minutes.
After the cheesecake has cooled, refrigerate it for at least 4 hours, or ideally overnight, before removing it from the springform pan.
Before serving, top the cheesecake with the berry topping.