Position a rack in the lower third of the oven and preheat to 325 degrees . Lightly coat a 9-inch springform pan with cooking spray. Using a food processor, grind the cookies with 1/4 cup sugar. Add the butter and pulse to combine. Press the mixture into the bottom and halfway up the sides of the prepared pan. Bake until firm, 3 minutes. Transfer to a rack to cool.
Using an electric mixer, beat the cream cheese and remaining 1-1/2 cups sugar until smooth, about 1 minute. Beat in 2 eggs until blended. Transfer 1/4 cup of the cream cheese mixture to a small bowl and set aside. Add the remaining 3 eggs, the pumpkin, vanilla, pumpkin-pie spice and salt and beat until combined, 2 minutes.
Set the springform pan with the cooled crust on a double layer of heavy-duty foil and wrap the foil tightly around the bottom and sides; set in a roasting pan. Pour the cheesecake filling into the crust. Dollop the reserved cream cheese mixture onto the filling and, using the tip of a knife, swirl together. Transfer the roasting pan to the oven and fill with enough boiling water to reach about halfway up the sides of the springform pan. Bake the cheesecake until the center is slightly wobbly but the edges are firm to the touch, about 1-1/4 hours.
Transfer the roasting pan with the cheesecake to a rack and let cool for 45 minutes. Remove the springform pan, discard the foil and let the cheesecake cool on the rack for 3 hours. Run a knife around the edges to loosen the cake. Wrap the pan in plastic wrap; refrigerate overnight.
Remove the springform pan sides. Run a knife under the bottom of the cake to release it, then slide onto a serving platter.