Finely chop the pistachios using a food processor or a sharp knife. Leave a few slightly larger pieces for texture if desired.
In a large mixing bowl, stir together the chopped pistachios, flour, 1/4 cup (33g) of the powdered sugar, and the salt.
Add the softened butter and vanilla extract to the dry mixture. Use your hands or a sturdy spoon to mix and mash the ingredients until a soft dough forms.
Cover the bowl and place it in the refrigerator to chill for 30 minutes.
Preheat the oven to 350°F (175°C). Line a large baking sheet with parchment paper.
Scoop 1-tablespoon portions of the chilled dough and roll them into smooth balls. Place them on the prepared baking sheet with about 1 inch of space between each ball.
Bake for 12 to 15 minutes, until the bottom edges turn light brown.
Let the cookies cool on the baking sheet for 5 minutes. Place the remaining 3/4 cup (97g) of powdered sugar in a medium bowl and gently toss the warm cookies in the sugar to coat them.
Transfer the coated cookies to a wire rack to cool completely. Toss them in the powdered sugar a second time once they reach room temperature.