Beat butter and ½ cup of the powdered sugar in a stand mixer fitted with a paddle attachment on medium speed until smooth, about 1 minute. Beat in vanilla and chopped pistachios until combined, about 30 seconds.

Whisk together flour and salt in a medium bowl until combined. With mixer on low speed, gradually add flour mixture to butter mixture, beating just until dough comes together, about 1 minute, stopping to scrape down sides of bowl as needed. Shape dough into a 6-inch disk, and wrap in plastic wrap. Let chill until firm, at least 1 hour or up to 24 hours.

Preheat oven to 350°F. Line a large rimmed baking sheet with parchment paper. Roll out dough on a lightly floured work surface to ¼-inch thickness. Cut desired shapes using lightly floured 2-inch cookie cutters. Chill, reroll, and cut dough scraps as needed. Transfer shapes to prepared baking sheet, spacing 1 inch apart. Freeze until firm, about 10 minutes.

Bake in preheated oven until the edges are set, 14 to 16 minutes. Let cool on baking sheet for 5 minutes. Transfer cookies to a wire rack; let cool completely, about 30 minutes.

In a medium bowl, whisk together lemon juice and remaining 1 cup powdered sugar until smooth. Dip, drizzle, or spread glaze over cooled cookies as desired. Crush additional chopped pistachios; sprinkle over cookies. Let stand until glaze is set, about 30 minutes.