In a food processor, blend the graham crackers with light brown sugar and melted butter until the mixture resembles wet sand. Press this mixture firmly into the bottom of a 9-inch springform pan, ensuring it's evenly distributed. Chill the crust in the freezer for about 15 minutes.

In a large mixing bowl, beat the softened cream cheese and light brown sugar together until smooth and creamy, about 2-3 minutes. Mix in the sour cream, lemon juice, and vanilla extract, blending until fully incorporated. Fold in the frozen whipped topping gently.

Carefully pour the cream cheese filling over the chilled graham cracker crust. Spread it out evenly with a spatula. Cover the springform pan with plastic wrap and refrigerate for at least 4 to 6 hours, or preferably overnight.

In a saucepan, combine the blackberries, additional light brown sugar, and water. Cook over medium heat for about 3 minutes, gently pressing the blackberries to release their juices.

Mix the cornstarch with a splash of lemon juice until smooth. Stir this into the blackberry sauce in the saucepan and continue cooking until it thickens.

Allow the blackberry topping to cool at room temperature, then refrigerate until ready to assemble your cheesecake.

Once the cheesecake has set, release the sides of the springform pan. Spoon the cooled blackberry topping over the cheesecake.