Make the crust: Position a rack in the lower third of the oven; preheat to 325˚. Spread the whole and chopped pistachios on a rimmed baking sheet (keep them separate) and bake until toasted, about 10 minutes. Let cool.
Set aside the chopped pistachios in a small bowl. Pulse the toasted whole pistachios in a food processor along with the shortbread cookies and granulated sugar until finely ground. Add the melted butter and pulse until the mixture looks like wet sand. Press the mixture into the bottom of an 8-inch round springform pan. Bake until set around the edges and no longer puffed, about 20 minutes. Transfer to a rack; let cool 10 minutes.
Meanwhile, make the filling: Beat the eggs and granulated sugar in a large bowl with a mixer on medium-high speed until pale and thickened, about 5 minutes. Add the ricotta, pistachio paste, lemon zest, vanilla and almond extracts and salt and continue beating until smooth, about 2 minutes. Add the green food coloring and beat until combined.
Pour the filling over the crust. Bake until the cheesecake is puffed and the edges are set but the center still jiggles slightly, about 1 1/2 hours.
Transfer the cheesecake to a rack and let cool to room temperature, 2 to 3 hours. Run a thin knife around the edge of the cheesecake to loosen, then remove the springform ring. Serve at room temperature or cover and refrigerate until chilled, at least 2 hours or overnight. Dust with confectioners' sugar and sprinkle with the toasted chopped pistachios before serving.