Measure 1/4 cup (60 ml) of the pistachios and set them aside in a small bowl. Place the flour, remaining pistachios, and lemon zest in a food processor. Blend until the pistachios are finely ground. Transfer to a bowl, then whisk in the Parmigiano-Reggiano cheese and baking powder. Set aside.
In the bowl of a stand mixer, or in a large mixing bowl if using a hand mixer, add the eggs and sugar and beat until fluffy and pale, about 2 minutes. Mix in the oil, then add the dry ingredients and mix until just combined.
Pour the Parmesan Pistachio Cake batter into the prepared pan. Coarsely chop the saved 1/4 cup (60 ml) pistachios and sprinkle over the cake. Bake for about 30 minutes, or until the cake is puffed and golden and a toothpick inserted in the center of the cake comes out clean.
Transfer the cake to a cooling rack and let cool to room temperature. Once cool, wrap tightly in plastic and let rest at room temperature overnight. The cake is tasty right away, but the flavors will deepen and improve as the cake rests for a few hours.
For the Basil-Infused Stewed Apricots: In a large skillet set over medium-high heat, whisk together the sugar, water, and lemon juice. Simmer until the sugar is fully melted. Lay the apricot halves side-by-side, cut-side down, into the syrup. Simmer for 3 minutes. Turn each of the apricots, push the basil into the syrup, and simmer for an additional 2 minutes, or until the apricots are tender but not breaking down. (Decrease the cooking times if the apricots you’re using are very ripe.) Remove from the heat, let the stewed apricots cool completely in the skillet, then transfer them with the syrup and basil leaves to a glass jar or airtight container.
TO SERVE: Serve the Parmesan Pistachio Cake at room temperature accompanied by room temperature or slightly reheated Basil-Infused Stewed Apricots and Vanilla Bean Gelato, if desired.
STORAGE: The Parmesan Pistachio Cake will keep, wrapped in plastic, in an airtight container, or under a cake dome, at room temperature for up to 3 days. The Basil-Infused Stewed Apricots will keep, refrigerated, for up to 3 days.
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