Preheat the oven to 325°F (160°C) and line a 12-cup muffin tin with paper liners.

1. Make the crust: In a medium bowl, mix together the graham cracker crumbs, sugar, and melted butter until the mixture resembles wet sand. Spoon about 1 tablespoon of the mixture into each muffin cup and press down firmly using the back of a spoon or your fingers. Bake for 8-10 minutes, until golden and set. Remove from the oven and let cool while you prepare the cheesecake filling.
2. Prepare the cheesecake filling: In a large mixing bowl, beat the softened cream cheese and sugar together until smooth and creamy. Add the egg and vanilla extract, and beat until fully incorporated. Finally, mix in the sour cream until the filling is smooth and light.
3. Assemble the cheesecakes: Spoon or pipe the cheesecake filling over the cooled graham cracker crusts in each muffin cup, filling them nearly to the top. Smooth the tops with a spatula.
4. Bake the cheesecakes: Bake for 18-20 minutes, or until the cheesecakes are set around the edges and slightly jiggly in the center. Be careful not to overbake, as the center should remain soft. Allow the cheesecakes to cool at room temperature for 10 minutes, then transfer to the fridge to chill for at least 2 hours (or overnight) to set completely.
5. Make the blackberry topping: While the cheesecakes chill, prepare the blackberry topping. In a small saucepan, combine the blackberries, sugar, and lemon juice over medium heat. Cook, stirring occasionally, until the blackberries begin to break down and release their juices, about 5-7 minutes. Once the mixture has thickened slightly, remove it from the heat and let it cool.
6. Top and serve: Once the cheesecakes have fully set, spoon the blackberry topping over each mini cheesecake just before serving. Garnish with additional fresh blackberries if desired.