For the crust: Position a rack in the middle of the oven and preheat to 325 degrees F.
Add the cookies and sugar to the bowl of a food processor and pulse to fine crumbs. Drizzle in the butter and pulse until the crumbs are completely coated with the butter.
Press the crumb mixture firmly into the bottom of a 10-inch springform pan. Bake until set, 15 to 18 minutes; let cool completely on a rack, 30 minutes.
Wrap the bottom and sides of the pan with a large piece of foil and put it in a large roasting pan.
For the filling: Beat together the cream cheese and sugar in a large bowl with an electric mixer on medium speed for 1 minute. Add the sour cream and beat until just combined. Add the heavy cream and beat until just combined. Mix in the eggs by hand, one at a time, until just combined. Mix in the vanilla by hand until just combined. (Take care not to overmix or the cheesecake will turn into a souffle!). Set aside.
Puree half the blackberries in a blender until completely smooth. Transfer to a small bowl and ladle in 1/4 cup of the cheesecake mixture. Stir to combine. Pour the remaining plain cheesecake batter into the crust.
Use an ice cream scoop or large spoon to drop scoopfuls of the blackberry batter on top of the plain batter. Drag a long wooden skewer through the batter, making sure the skewer reaches down to the bottom, to marble the colors for a tie-dye effect. Add enough hot water to the roasting pan to come about halfway up the sides of the springform pan.
Bake until the outside of the cheesecake is set, but the center is still slightly loose, about 1 hour 20 minutes. Turn the oven off and leave the cheesecake inside for 1 hour.
Remove the cheesecake from the roasting pan to a cooling rack. Run a knife around the edge of the pan and let the cheesecake cool to room temperature, about 2 hours. Cover and refrigerate at least 8 hours and up to 12 hours.
Run a knife around the edge of the pan one more time, then open the pan clasp to release the sides. Transfer the cheesecake to a serving platter or cake stand. Smooth the edges of the cheesecake with a knife. Slice and serve with whipped cream and the remaining blackberries.
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