Preheat oven to 350°F (175°C). Grease a 9-inch round springform pan.
Combine the cookie crumbs and butter until thoroughly mixed.
Press the crust mixture into the bottom of the prepared pan.
Place the pan on a lined, rimmed baking sheet to catch any spills. Bake for 10 minutes. Set aside.
To make the cheesecake:
Using an electric mixer on medium speed, beat the cream cheese and brown sugar until thoroughly combined. Add the eggs and vanilla, and mix well. Mix in the sour cream.
Pour about half of the cheesecake filling over the cooled crust, and spread evenly.
Dollop small amounts of about half of the apple butter over the filling. Use a thin knife or spatula to swirl gently into the filling.
Add the remaining filling to the pan and spread evenly. (You may find it more effective to drop the filling over the top to minimize how much you'll need to spread it, as that may negate some of the swirling.)
Dollop the remaining apple butter over the top of the filling. Use a thin knife or spatula to swirl gently into the filling.
Place the pan on a lined, rimmed baking sheet to catch any spills. Bake for 10 minutes.
Reduce oven temperature to 200°F (95°C). Bake for 2 hours, or until the top seems almost set.
Remove the cheesecake from the oven, and carefully run a thin knife around the edges of the pan.
Turn off the oven, and return the cheesecake to the oven. Leave the cheesecake in the oven for 2 hours.