Special equipment: a 10-inch springform pan, two 4-ounce ladles
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. Cool completely on a rack, about 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 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!) Remove 5 cups of the batter to a large bowl and add 1/4 cup water. Stir until smooth and set aside.
Add the cocoa powder and 1 cup water to the bowl with the remaining batter. Beat with an electric mixer on low speed until well combined and smooth, about 1 minute.
Use a 4-ounce (1/4-cup) ladle to pour some of the chocolate batter into the center of the crust. Use a second 4-ounce (1/4-cup) ladle to pour the white batter into the middle of the chocolate batter. (The batters will spread and as you add alternating scoops of batter, they will widen forming concentric circles.) Add a ladle of chocolate batter to the center of the white batter. Repeat with the remaining batters, alternating between the flavors until all the batter is gone. 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.
Transfer 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.
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