Step 1: Preheat your oven to 350°F (175°C).
Step 2: In a medium bowl, combine the graham cracker crumbs and granulated sugar.
Step 3: Pour in the melted butter and mix until the crumbs are evenly moistened.
Step 4: Press the mixture firmly into the bottom of a 9-inch springform pan. You can use the bottom of a glass or a flat measuring cup to get an even layer.
Step 5: Bake for 8-10 minutes, then remove from the oven and let cool slightly while you prepare the filling.
Step 6: In a small saucepan, combine the mixed berries, granulated sugar, and lemon juice.
Step 7: Cook over medium heat, stirring occasionally, until the berries break down and the mixture starts to simmer, about 5-7 minutes.
Step 8: In a small bowl, whisk together the cornstarch and cold water to create a slurry.
Step 9: Pour the slurry into the simmering berry mixture, stirring constantly. Continue to cook for another 1-2 minutes, until the sauce thickens.
Step 10: Remove from heat and let cool completely. Once cooled, lightly mash with a fork for a chunkier swirl, or press through a fine-mesh sieve for a smoother sauce.
Step 11: Reduce oven temperature to 325°F (160°C). Prepare a water bath by wrapping the outside of your springform pan tightly with several layers of heavy-duty aluminum foil to prevent water from seeping in. Place the foiled pan into a larger roasting pan.
Step 12: In a large mixing bowl, using an electric mixer on medium speed, beat the softened cream cheese until smooth and creamy, about 2-3 minutes. Scrape down the sides of the bowl.
Step 13: Gradually add the granulated sugar, beating until just combined and smooth. Do not overmix.
Step 14: Add the all-purpose flour, vanilla extract, and lemon zest. Beat on low speed until just incorporated.
Step 15: Add the eggs one at a time, beating on low speed after each addition just until the yolk disappears. Be careful not to incorporate too much air into the batter, as this can cause cracks.
Step 16: Gently fold in the room temperature sour cream and a pinch of salt until just combined.
Step 17: Pour about two-thirds of the cheesecake batter over the cooled crust in the springform pan.
Step 18: Spoon half of the cooled berry swirl mixture in dollops over the batter.
Step 19: Carefully pour the remaining cheesecake batter over the berry mixture.
Step 20: Spoon the remaining berry swirl mixture over the top. Using a butter knife or skewer, gently swirl the berry mixture into the cheesecake batter to create beautiful patterns.
Step 21: Carefully pour hot water into the roasting pan around the springform pan, ensuring the water comes up about halfway up the sides of the cheesecake pan.
Step 22: Bake for 1 hour to 1 hour 15 minutes, or until the edges are set but the center still jiggles slightly when gently shaken.
Step 23: Turn off the oven, crack the door open slightly, and let the cheesecake cool in the water bath for 1 hour.
Step 24: Remove the cheesecake from the water bath and the roasting pan. Carefully remove the foil. Run a thin knife around the edge of the pan to loosen the cheesecake.
Step 25: Let cool completely at room temperature, then refrigerate for at least 6 hours, or preferably overnight, before serving.