Prepare the Crust:

1. Preheat the oven to 325°F (160°C). Line a 12-cup muffin pan with cupcake liners.
2. In a bowl, mix graham cracker crumbs, sugar, and melted butter until evenly combined.
3. Press about 1 tablespoon of the crust mixture into the bottom of each liner. Press firmly with the back of a spoon or your fingers.
4. Bake for 5 minutes, then set aside to cool.

2. Make the Cheesecake Filling:

1. In a large mixing bowl, beat cream cheese and sugar together until smooth.
2. Add eggs one at a time, mixing well after each addition.
3. Stir in sour cream, lime juice, lime zest, and vanilla extract until creamy and well combined.
4. Divide the filling evenly among the cupcake liners, filling each about 3/4 full.

3. Bake the Cheesecakes:

1. Bake for 18–22 minutes, or until the centers are slightly jiggly but set.
2. Remove from the oven and let cool to room temperature in the pan. Then refrigerate for at least 2 hours, or until fully chilled.

4. Prepare the Blackberry Topping:

1. In a small saucepan, combine blackberries, sugar, cornstarch, water, and lime juice.
2. Cook over medium heat, stirring frequently, until the mixture thickens and the berries start to break down (about 5–7 minutes).
3. Let the topping cool completely.

5. Assemble the Cupcakes:

1. Spoon the blackberry topping over each chilled cheesecake cupcake.
2. Garnish with a blackberry or a small sprinkle of lime zest, if desired.