In a food processor, pulse together the graham crackers, light brown sugar, and melted salted butter until the mixture resembles wet sand. Firmly press this combination into the bottom and up the sides of a 9-inch springform pan for a sturdy crust. Refrigerate for at least 30 minutes to allow the crust to set before adding the filling.

In a large mixing bowl, blend the room-temperature cream cheese, light brown sugar, sour cream, lemon juice, and vanilla extract until smooth and creamy. Gently fold in the thawed frozen whipped topping.

Carefully pour the filling into the chilled crust, using a spatula to smooth the top. Cover the cheesecake with plastic wrap and refrigerate for at least 4 to 6 hours.

In a medium saucepan over medium heat, combine the blackberries, light brown sugar, and water. Cook for about 5-7 minutes until the berries soften. Slowly add the cornstarch slurry to thicken the sauce.

Remove the blackberry mixture from heat and let it cool at room temperature before transferring it to the refrigerator for about 30 minutes.

Once the cheesecake has set and the topping is cooled, gently remove the sides of the springform pan. Pour the cooled blackberry topping evenly over the cheesecake, slice and serve chilled.