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Butter the sides of a 9-inch springform pan and coat the butter with some of the graham cracker crumbs
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Melt the remaining butter in a saucepan or microwave in a microwave-safe dish.
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Tossing with a fork, combine the remaining graham cracker crumbs and the melted butter.
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Press the crumb mixture into the bottom of the springform pan, and try not to disturb the crumbs on the side.
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Wrap the underside of the pan with one sheet of foil to create an extra seal from the water bath. Press the foil against the sides.
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To make the filling.
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Add the ¾ cup of cream to a microwave dish or second saucepan. Split the vanilla bean and scrape the seeds, as well as place the pods into the cream.
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Heat the cream until it barely simmers, take off the heat and then let it steep for 15 minutes before letting it cool completely.
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Preheat the oven to 350°
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Place a kettle or large pot on the stove to heat some boiling water and have ready a large baking pan big enough to fit a 9-inch springform pan in for a water bath.
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Place ½ lb. of the cream cheese (1 8 oz package), 1/3 cup of the sugar, and the cornstarch in a large bowl.
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Beat with an electric mixer on low until creamy, about 3 minutes, then beat in the remaining 3 packages of the cream cheese.
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Increase the mixer speed to high and beat in the remaining 1 1/3 cups of the sugar, then beat in the vanilla bean paste if not using the vanilla bean.
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Blend in the eggs, one at a time, beating the batter well after each addition.
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Remove the vanilla bean pod but leave the seeds in the cream. Reserve the vanilla bean pod for the peach topping.
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Blend in the heavy cream that has been infused with the vanilla.
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At this point, mix the filling only until thoroughly blended. Be careful not to overmix the batter.
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Gently spoon the filling onto the prepared crust.
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Place the springform pan in a large shallow pan containing hot water that comes about 1 inch up the side of the pan.
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Bake the cheesecake until the center barely jiggles when you shake the pan, about 1 hour.
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Cool the cake on a wire rack for 1 hour.
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Then cover the cake with plastic wrap and refrigerate until it's completely cold, at least 4 hours or overnight.
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To make the peach topping.
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Pit the peaches and slice ½ inch thick
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Add the peaches along with the butter, reserved vanilla bean if using, or the paste if not, water, sugar, brown sugar, and cornstarch to a saucepan. Heat to a simmer and cook until the peaches have released their moisture and the mixture has thickened, stirring occasionally. You still want to have slices of peaches, so don't cook them to jam.
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Let cool, remove the vanilla bean pod, if using, and then chill until ready to top the cheesecake.
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When ready to serve the cheesecake.
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Remove the sides of the springform pan.