Special equipment: A 9-inch springform pan
Position an oven rack to the bottom third of the oven and preheat the oven to 325 degrees F.
Spread the pistachios out on a baking sheet and bake until golden and toasted, 10 to 12 minutes. Transfer to a heat-safe plate to cool, then pulse them in a food processor until very finely ground. Transfer the ground pistachios to a small bowl and set aside.
Set a 9-inch springform pan on top of a piece of parchment paper and trace a circle around it. Cut out the circle and set aside. Generously grease the bottom and sides of the pan with the room temperature butter, then press the parchment circle into the bottom of the pan.
Put the ricotta and 2/3 cup sugar in the bowl of food processor and process until the mixture is smooth, about 30 seconds. Transfer the ricotta mixture to a large bowl and whisk in the eggs yolks, orange zest and juice and the vanilla.
In another large bowl, add the egg whites and whip using an electric mixer on medium-high speed until the eggs become foamy. Sprinkle in the remaining 2 tablespoons of sugar and continue to whip the egg whites until medium peaks form, about 2 minutes. Use a rubber spatula to fold 1/3 of the beaten whites into the ricotta mixture until just incorporated and a few white streaks remain. Gently fold in the remaining whipped egg whites, folding until there are no white streaks remaining.
Pour the mixture into the prepared springform pan and set it on a baking sheet. Bake until the cheesecake is light golden brown and the sides begin to pull away from the pan, 55 minutes to about 1 hour and 5 minutes. Transfer the cheesecake to a cooling rack and let it cool completely, about 1 hour. Cover the pan with plastic wrap and refrigerate at least 2 hours and up to 2 days before serving.
To serve, run a thin knife or spatula around the cake to release it from the sides, then unclasp the latch of the pan. Dust the top of the cake with confectioners' sugar and sprinkle with the finely ground pistachios. Set the cheesecake on a cake platter, slice and serve.
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