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Preheat the oven to 350°F. Lightly grease a 9" round springform pan or deep 9" round removable-bottom pan. This cake is very tall, and requires an extra-deep pan. Measure your pan; if it's not at least 2 3/4" deep, don't attempt this recipe. Wrap the bottom and sides of the pan with aluminum foil, preferably a single sheet.
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To make the crust: Place the flour, 2 tablespoons of the sugar, the baking powder, salt, vegetable oil or butter, 3 egg yolks, and vanilla into a large mixing bowl. Beat until well combined; the mixture will be stiff and somewhat crumbly.
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In a separate bowl, beat the egg whites with the cream of tartar until they're frothy. Add the remaining 4 tablespoons of sugar gradually, beating all the while, until the mixture is stiff and glossy.
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Working in thirds, mix the beaten egg whites into the batter. Take care to keep the batter light; mix gently, don't beat. You may find at the end there are still some tiny lumps in the batter; that's OK.
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Spoon the batter into the prepared pan. Bake for about 15 minutes, until the cake has risen, is barely beginning to brown, and a toothpick or thin paring knife inserted into the center comes out clean.
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Remove the cake from the oven, and immediately loosen the edges with a table knife or thin spatula. Allow it to cool in the pan while you make the filling. It'll settle and shrink a bit as it cools; that's OK. Leave the oven on.
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To make the filling: Place 8 ounces (1 package) of the room-temperature cream cheese, 1/3 cup of the sugar, and the cornstarch in a mixing bowl. Mix on low speed until smooth.
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Add the remaining cream cheese 8 ounces at a time, beating on low speed until smooth.
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Slowly beat in the remaining 1 1/3 cups sugar, and the vanilla.
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Add the eggs one at a time, beating well after each addition.
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Add the cream, beating just until blended.
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Spoon the batter over the cake in the pan. The filling will expand and rise, so make sure you don't fill the pan right to the brim.
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Place the pan into a larger pan, and fill the larger pan with enough hot water to come 1" up the sides of the springform pan. Place both pans on a lower-middle rack of your oven.
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Bake the cheesecake for 75 to 90 minutes, until the cake is just barely beginning to turn golden around the edges, and the top appears set. The center will still look jiggly; that's OK. A thermometer inserted into the center should register about 160°F to 165°F.
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Remove the cake from the oven, and gently lift it out of the water bath onto a rack. Allow it to cool at room temperature, undisturbed, for 2 to 3 hours, until it's no longer warm to the touch.
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Refrigerate the cake, covered, until you're ready to serve it.
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To serve, slice with a knife dipped in hot water and wiped dry. Repeat this step after every slice. This cake is traditionally served without topping; but feel free to add your own favorite, if desired.
