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MAKE THE CAKE:.
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Preheat the oven to 350F.
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Butter a 10-inch round cake pan, line the bottom with parchment paper and butter the paper.
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In a food processor, pulse the almonds with 2/3 cup of the pistachios until finely ground.
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In a large bowl, using a handheld electric mixer, beat the butter with the sugar at medium speed until its the texture of moist sand.
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Add the eggs one at a time, beating well between additions.
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Add the lemon zest, lemon juice and vanilla seeds and beat until smooth.
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Gradually beat in the olive oil in a thin stream.
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Using a rubber spatula, fold in the ground nuts, flour and salt.
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Scrape the batter into the prepared pan.
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Bake the cake for about 55 minutes, or until a toothpick inserted in the center comes out clean.
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Transfer the pan to a rack and let the cake cool for 15 minutes.
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Invert the cake onto the rack and peel off the parchment paper.
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Invert the cake again onto a serving plate.
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Meanwhile, toast the remaining 1/2 cup of pistachios in a pie plate for about 8 minutes, or until fragrant.
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MAKE THE SYRUP:.
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In a small saucepan, combine the lemon zest, lemon juice, sugar and water and bring to a simmer over moderate heat, stirring until the sugar is dissolved.
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Pour the syrup over the warm cake and scatter the toasted pistachios all over the top.
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Let stand for 1 hour, then cut into wedges and serve with creme fraiche.
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**The cake can be kept in an airtight container at room temperature for up to 2 days.
