Preheat oven to 350 degrees. Place cookie crumbs, walnuts, and 3 tablespoons sugar in a food processor and pulse 3 or 4 times until combined; add butter and pulse 4 or 5 times, or until crumbly. Wrap the outside of a 10-inch springform pan with aluminum foil, then press cookie mixture into the bottom and 1 inch up the sides. Bake 8 to 10 minutes. Cool crust completely on a wire rack. Reduce oven heat to 325 degrees.
Beat cream cheese with an electric mixer at medium speed 3 minutes, or until creamy. Gradually add brown sugar and the remaining 3/4 cup sugar, beating 4 minutes, or until blended.
Add eggs, one at a time, beating just until yellow disappears. Add pumpkin, whipping cream, and pumpkin-pie spice. Beat on low speed of an electric mixer just until blended. Do not overbeat. Pour batter over prepared crust.
Bake 1 hour and 15 minutes, or until almost set (it will still jiggle in the center). Remove from oven; run a knife around edge of pan. Cool on a wire rack 1 hour. Chill 8 hours or overnight. Garnish with walnut halves and serve with Caramel Sauce.
Caramel Sauce: Melt butter in a heavy saucepan over medium heat; add brown sugar and corn syrup, then cook, stirring constantly, until mixture comes to a boil. Gradually add cream, stirring constantly. Cook, stirring constantly, until mixture comes to a boil. Remove from heat. Cool to room temperature.