Adjust an oven rack to the middle of the oven. Preheat the oven to 350F [180C]. Line two baking sheets with parchment paper.
Place the pistachios in a food processor fitted with a steel blade and pulse until finely ground. Add the flour, baking soda, and salt. Pulse to fully combine and set aside.
In the bowl of a stand mixer fitted with a paddle, beat the butter on medium until creamy. Add the granulated and confectioners' sugars and beat on medium until light and fluffy, 2 to 3 minutes. Add the olive oil and mix on low until combined. Scrape down the sides of the bowl and add the egg and vanilla, mixing on low until combined. Add the flour mixture and mix on low until combined.
Gather the dough, wrap with plastic wrap, and chill in the refrigerator for at least 2 hours and up to 1 day.
Lightly flour a work surface and roll the dough to 1/4 in [6 mm] thick. Using a 2 in [12 cm] biscuit or cookie cutter, cut out circles. (Any dough scraps can be rewrapped and chilled while the cookies are baking.) Gently slide a metal spatula underneath each round and transfer it to the prepared baking sheet.
Place 12 cookies on each sheet. Put the first baking sheet in the freezer for 10 minutes. After the dough has chilled, put the first pan of cookies in the oven, and then put the second pan in the freezer. Repeat with the leftover dough.
Bake one sheet at a time, 10 to 12 minutes, until the cookies are just beginning to brown on the edges.
Transfer the baking sheet to a wire rack and let the cookies cool completely on the pan.