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Preheat the oven to 350u00b0F. Prepare a 9-inch springform pan by spraying the inside with baking spray and wrapping the outside with a double layer of heavy-duty foil; set aside.
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For the crust:
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In a food processor, process the cookies with the pecans, sugar, salt and cinnamon until the cookies become fine crumbs, about 20 seconds. Add the melted butter and pulse until combined. Press mixture into the base of the prepared pan and part way up the sides. Cover the crust with plastic wrap and refrigerate until needed.
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For the filling:
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In a small heavy-bottomed saucepan, stir the pumpkin puree and sugar together over medium heat. Bring the mixture to a sputtering simmer, stirring constantly for about 5 minutes. Turn the heat to low and cook, stirring constantly, until the mixture has darkened and thickened to the consistency of applesauce, about 5 minutes longer.
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Scrape the mixture into a large food processor and process for 1 minute with the feed tube open (so steam can escape), scraping down the sides when needed. With the motor running, add the chilled cream and softened cream cheese and process for 30 seconds or until smooth and incorporated, scraping down the sides of the bowl two or three times. Add the eggs and yolks and process for about 5 seconds. Pour into prepared pan and smooth the top.
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Set the springform pan into a larger pan (a 12x2-inch cake pan or a roasting pan) and pour about 1 inch of very hot water into the outer pan. Carefully move cheesecake (still inside of the water filled pan) to the oven and bake for 45 minutes. Then turn off the oven without opening the door and let the cheesecake cool for 1 hour. Then transfer the cake to a rack (the center will still be jiggly) and cool to room temperature, about 1 hour. Cover the pan with plastic wrap and refrigerate for at least 6 hours or overnight.
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For the caramel sauce:
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In a medium heavy-bottom saucepan, stir together the sugar, syrup and water until the sugar is completely moistened. Heat over medium heat, stirring constantly, until the sugar dissolves and the syrup is bubbling. Then stop stirring completely and allow it to boil undisturbed until it turns a deep amber color (360u00b0F on a candy thermometer). Immediately remove pan from the heat and slowly and carefully pour the hot cream into the caramel. It will bubble up furiously so be careful.
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Use a high-temperature spatula to stir the mixture until smooth, scraping the thicker part that settled on the bottom. If any lumps develop, return the pan to the heat and stir until they dissolve. Off the heat, stir in the butter. The mixture will be streaky but will become uniform after cooling slightly and stirring. Allow the caramel to cool for 3 minutes, then gently stir in the vanilla extract. Cool to room temperature before drizzling on cheesecake. Garnish the cheesecake with whipped cream and pecan halves as desired before slicing and serving.
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Recipe adapted from Rose Levy Beranbaum.
