MAKE THE CAKE: Preheat the oven to 350 degrees. Butter a 10-inch round cake pan, line the bottom with parchment paper and butter the paper. In a food processor, pulse the almonds with 2/3 cup of the pistachios until finely ground.

In a large bowl, using a handheld electric mixer, beat the butter with the sugar at medium speed until it's the texture of moist sand. Add the eggs one at a time, beating well between additions. Add the lemon zest, lemon juice, and vanilla seeds and beat until smooth. Gradually beat in the olive oil in a thin stream. Using a rubber spatula, fold in the ground nuts, flour, and salt. Scrape the batter into the prepared pan.

Bake the cake for about 55 minutes, or until a toothpick inserted in the center comes out clean. Transfer the pan to a rack and let the cake cool for 15 minutes. Invert the cake onto the rack and peel off the parchment paper. Invert the cake again onto a serving plate.

Meanwhile, toast the remaining 1/2 cup of pistachios in a pie plate for about 8 minutes, or until fragrant.

MAKE THE SYRUP: 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. Pour the syrup over the warm cake and scatter the toasted pistachios all over the top. Let stand for 1 hour, then cut into wedges and serve with creme fraiche.

The cake can be kept in an airtight container at room temperature for up to 2 days.