Position the oven racks in the center and lower third of the oven. Fill a baking dish or roasting pan about half full with water and place on the lower rack. Preheat the oven to 325 degrees F. Brush the bottom and sides of a 9-inch springform pan with butter and set aside.
Grind the graham crackers and 1 tablespoon of the granulated sugar in a food processor until fine crumbs. Add the butter and 1 tablespoon water and pulse until moistened. Press into the bottom of the prepared pan. Bake on the center rack until firm and browned, 10 to 12 minutes. Let cool completely.
Beat the cream cheese, brown sugar, salt and remaining 1/3 cup granulated sugar with an electric mixer on medium speed in a large bowl until smooth and fluffy. Reduce the mixer speed to low and beat in the yogurt and eggs until blended. Then beat in the pumpkin puree, flour, pie spice and vanilla until just combined. Stir with a spatula a few times to make sure all the ingredients are incorporated from the bottom of the bowl.
Pour the batter over the crust and rap the pan against the countertop to release air bubbles. Bake the cheesecake on the center rack until the center is only slightly jiggly, 50 to 60 minutes. Turn off the oven and crack the door open. Let the cheesecake sit in the oven for 15 minutes, then transfer it to a wire rack. Run a thin sharp knife between the cheesecake and the pan to release the sides. Let cool completely, then refrigerate until cold, at least 4 hours and up to overnight. Release the sides of the pan and dust the top with the confectioners' sugar before serving.
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