How to Make It
In a medium saucepan, combine thawed strawberries and any juices with 2/3 cup sugar and 2 tablespoons lemon juice. Cook over medium heat, stirring occasionally until the sugar dissolves. Bring to a boil and cook until the berries are soft and starting to dissolve and fall apart, about 8 to 10 minutes. Remove from heat and puree until smooth with an immersion blender. (Alternatively, cool completely and puree in a food processor or traditional blender.) Transfer puree to a small bowl to cool completely.
Simultaneously, in a second saucepan, combine thawed blueberries and any juices with 3 tablespoons sugar and 1 tablespoon lemon juice. Cook over medium heat, stirring occasionally, until the sugar dissolves and the berries start to soften. Cover the pan and cook until the berries are soft enough to gently crush with the back of a spoon, about 8 minutes. Remove from heat, cool slightly, and puree with an immersion blender or in a food processor or traditional blender. For a smoother puree, press the blueberry sauce through a fine mesh sieve. Transfer puree to a small bowl to cool completely.
Preheat oven to 350 degrees. Spray a 10- x 15-inch sheet pan with nonstick cooking spray.
In a large bowl, combine the graham cracker crumbs with the melted butter and sugar. Stir to moisten the crumbs thoroughly with the butter. Press the crumbs into the bottom of the pan in an even layer. Bake for 10 minutes. Transfer to wire rack to cool slightly.
FOR THE FILLING: In a food processor fitted with a metal chopping blade, combine the cream cheese, eggs, sour cream, sugar, and vanilla. Process until very smooth. Pour the filling over the crumb crust and spread evenly. Give the pan a few sharp raps on the counter to eliminate any air bubbles. Drop the fruit purees in large dollops all over the surface of the cheesecake batter. With a bamboo skewer, pull through the center of each dollop two or three times to form star burst-like patterns all over the surface of the cheesecake.
Reduce the oven temperature to 325 degrees.
Bake the cheesecake until it is set around the edges and just barely wobbly in the very center, about 30 to 35 minutes. Turn off the oven and prop the door open with the handle of a wooden spoon and cool cheesecake for about 20 minutes. Remove the cheesecake from the oven and cool completely on a wire rack. When completely cool, refrigerate until ready to serve, at least 3 to 4 hours but preferably overnight.