Grind the cooled pistachios in a food processor into a fine powder. Add the flour and salt and pulse to combine; set aside.
Beat the butter, granulated sugar, and lemon zest with an electric mixer on medium-high in a medium bowl until fluffy, 2 to 3 minutes. Beat in the vanilla. Reduce speed to low and gradually add the pistachio-flour mixture, mixing just until combined (do not overmix). Chill for 1 hour and up to 2 days.
Heat oven to 325° F with the racks in the upper and lower thirds. Line two baking sheets with parchment. Place the confectioners’ sugar in a small bowl. Roll level tablespoonfuls of the dough into balls and place on the prepared baking sheets, spacing them 2 inches apart.
Bake, rotating the sheets halfway through, until the edges are firm and pale golden brown, 16 to 18 minutes. Cool slightly on the baking sheet, then toss the warm cookies in confectioners’ sugar to coat; transfer to a wire rack to cool completely. Dust with additional confectioners’ sugar before serving.
Storage suggestion: Keep the cookies in an airtight container at room temperature for up to 1 week.
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