01 - Preheat oven to 325°F. Lightly grease a 9-inch springform pan with butter or cooking spray. Cut a circle of parchment paper to fit the pan bottom and place inside, greasing the parchment as well.
02 - In a medium bowl, combine crushed graham crackers, melted butter, and 2 tablespoons sugar. Mix until crumbs are evenly moistened. Press mixture firmly and evenly into the bottom of the prepared springform pan using the back of a measuring cup or flat-bottomed glass. Bake for 8 minutes until lightly golden. Remove from oven and set aside to cool while preparing filling.
03 - Combine blackberries, ¼ cup sugar, and lemon juice in a small saucepan over medium heat. Cook for 8-10 minutes, stirring frequently and mashing berries with a spoon until they break down and mixture thickens. Remove from heat and use an immersion blender to smooth the mixture. Pour through a fine-mesh sieve set over a small bowl, pressing with a spatula to extract as much puree as possible while removing seeds. Discard seeds and set puree aside to cool completely.
04 - In a large bowl, beat softened cream cheese and 1 cup sugar with an electric mixer on medium-low speed until completely smooth and creamy, scraping down bowl sides as needed, about 2 minutes. Add eggs one at a time, beating well after each addition and scraping bowl to ensure thorough incorporation. Mix in sour cream, vanilla extract, and flour until just combined—do not overmix.
05 - Pour cheesecake batter over the cooled crust, smoothing top with an offset spatula. Drop spoonfuls of cooled blackberry puree evenly over the batter surface. Use a skewer, thin knife, or toothpick to gently swirl puree through batter in a circular or marble pattern, being careful not to over-swirl or disturb the crust layer.
06 - Place springform pan inside a large roasting pan or baking dish. Carefully pour boiling water into the larger pan until it reaches halfway up the sides of the springform pan, creating a water bath. Bake at 325°F for 50-60 minutes until edges are set but center still has a slight wobble when gently shaken—the center should not be liquid.
07 - Turn off oven and crack the door open slightly using a wooden spoon. Leave cheesecake inside for 1 hour to cool slowly, which prevents cracking. Remove from oven and water bath. Run a thin knife around the edge to loosen from pan sides. Cool completely at room temperature for 1 hour, then refrigerate uncovered for 4 hours or preferably overnight until thoroughly chilled and set.
08 - Release springform ring carefully. Transfer cheesecake to a serving plate using a large spatula. Garnish generously with fresh blackberries, mint leaves, and dollops of whipped cream if desired. Slice with a sharp knife dipped in hot water, wiping clean between cuts for clean slices.