Step 1: Prepare the Crust

1. Preheat your oven to 325°F (163°C) and line a standard 12-cup muffin tin with cupcake liners.
2. In a mixing bowl, combine the graham cracker crumbs and sugar. Stir in the melted butter until the mixture resembles wet sand.
3. Spoon about 1 tablespoon of the crust mixture into each liner and press it down firmly using the back of a spoon or the bottom of a small glass.
4. Bake the crusts for 5 minutes, then remove from the oven and let cool while you prepare the filling.

Step 2: Make the Cheesecake Filling

1. In a large mixing bowl, beat the softened cream cheese and sugar with an electric mixer on medium speed until smooth and creamy, about 2 minutes.
2. Add the eggs, one at a time, mixing well after each addition. Scrape down the sides of the bowl as needed.
3. Mix in the sour cream, vanilla extract, lime zest, and lime juice until fully incorporated and smooth.
4. Spoon the cheesecake filling evenly over the cooled crusts, filling each liner almost to the top.

Step 3: Bake the Cupcakes

1. Bake the cheesecake cupcakes in the preheated oven for 20-25 minutes, or until the centers are set and slightly jiggly. Avoid overbaking, as this can cause cracks.
2. Remove from the oven and let the cupcakes cool in the pan for 30 minutes. Transfer them to the refrigerator to chill for at least 2 hours, or until fully set.

Step 4: Prepare the Blackberry Lime Compote

1. In a small saucepan over medium heat, combine the blackberries, sugar, lime zest, and lime juice. Stir and cook until the blackberries release their juices and the mixture begins to simmer, about 5 minutes.
2. If desired, add the cornstarch slurry to thicken the compote. Simmer for an additional 2-3 minutes until thickened.
3. Remove from heat and let cool completely before spooning onto the chilled cupcakes.

Step 5: Assemble and Serve

1. Spoon a small amount of the blackberry lime compote onto each cheesecake cupcake.
2. Top with optional garnishes like whipped cream, fresh blackberries, or a sprinkle of lime zest.
3. Serve chilled and enjoy the zesty, fruity, and creamy combination!