Pulse the flour, granulated sugar, salt, pistachios, and vanilla in the bowl of a food processor for 1 to 2 minutes, until the pistachios are finely ground. Scatter the butter pieces in the food processor and pulse several times to cut the butter into the flour. Stop pulsing when the ingredients just come together and cling to one another.

Turn the dough onto a lightly floured work surface and shape it into a log about 2 inches in diameter and 10 inches long. Place the log on a piece of plastic wrap or parchment paper. Tightly roll the wrap around the log and twist the ends to seal them securely. Refrigerate for at least 2 hours or up to 5 days.

Position 2 racks in the center of the oven and preheat to 325ºF (163ºC). Line 2 rimmed baking sheets with parchment paper and set aside.

Cut the chilled cookie dough into ½-inch-thick coins and place about 2 inches apart on the prepared baking sheets. Lightly brush the top of each shortbread cookie with water and sprinkle with sugar. Bake for 15 to 18 minutes, or until the cookies are lightly golden brown on both top and bottom. Let the cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.

Store the pistachio shortbread cookies in an airtight container at room temperature for up to 2 weeks.