Preheat the oven to 275ºF and spray a 9-inch springform pan with nonstick cooking spray.
Make the Crust:
In a medium bowl, combine the cookie crumbs and butter. Transfer mixture to the prepared pan and press evenly into the bottom and up the sides. Chill crust in the refrigerator while you make the filling.
Make the Filling:
Place the butterscotch chips in a small bowl and heat in the microwave for one minute. Remove, stir, then microwave another 20-30 seconds until smooth. Set aside to cool slightly.*
In the bowl of a stand mixer fitted with the paddle attachment, beat the cream cheese and both sugars together until smooth and creamy, about 2-3 minutes. Add the Greek yogurt (or sour cream), eggs, vanilla, and salt and beat just until combined. Add the melted butterscotch chips and stir until smooth.
Pour cheesecake batter into the cooled crust, smoothing out the top. Bake for about 2 hours or until the sides of the cake are set (about 2 inches in) and the center still wiggles.
Once done, turn the oven off and keep the oven door open a few inches. Allow cheesecake to sit in the oven for 1 hour before transferring to a wire rack to cool. Refrigerate for at least 4 hours.
Make the Mousse:
In the bowl of a stand mixer fitted with the paddle attachment, beat the cream cheese, sugar, peanut butter, and vanilla together until smooth and creamy. Add 1/2 cup of the whipped topping and continue beating until lightened. Fold in the remaining 1 cup of whipped topping. Spread mousse evenly on top of completely cooled cheesecake, then refrigerate for an additional 2 hours, or until set.
To serve:
Pipe or dollop whipped cream, as desired. Garnish with additional Nutter Butter cookies. Run a knife around the edge of the pan, and remove the sides. Slice and serve.
Notes
*If melted butterscotch has already started to solidify before adding it to the batter, heat in the microwave at ten second intervals until smooth again.